I'd suggest a used road bike. In that price range a bike with an aluminium frame from the 2010s should be easy to find.

If you can buy a refurbished one from your local bike shop with some warranty that'd be the safest.

If you are not comfortable with drop bars my next suggestion would be a fitness/trekking bike, without any suspension.

Size fitting is important, check out any calculator what size would work for you.
Also you have to find well fitting contact points, especially when you start to go on longer rides: a comfortable saddle (harder is better in the long run), good grips/bar tape and pedals + shoes (clipless pedals later on perhaps)

As for training I don't think you can overdo it within sensible limits, your ass will hurt before that:).
You can check out interval training for building up stamina

In my opinion you should get a used steel frame road bike in good condition that's reasonably sized for you, and then take it to a bike shop to get it tuned up and to get a bike fit done. Once you have a season or two of riding under your belt then you could consider something newer and fancier and lighter.

The same rules apply about over training, ease into it, it will take time for your joints/ligaments/butt to adjust and there is no substitute for time in the saddle.

I'd go with an aluminum frame road bike if your intent is to work up to a 100 mile bike ride. Steel is more durable, but it is also way heavier which means over a longer ride, you are using more energy to keep things moving.

Why a road bike vs a hybrid or a mountain bike? Road bikes, hybrids, and mountain bikes all have different gear ratios, which impact your top speed. There are also weight differences between mountain bike frames, vs road bike frames, vs hybrids that also come into play. The rim/tire diameter impacts how far you travel per revolution. Even the way that tires are designed on these bikes make a difference.

-The gear ratio on a road bike can allow you to hit speeds of 50 mph while you are pedaling downhill. Depending on the mountain bike, your max speed might be closer to 20-30 miles per hour.

-Shocks on a mountain bike or hybrid will slightly rock as you pedal. This is stealing energy from you over long periods of time making your pedaling less efficient.

-Mountain bikes and sometimes hybrids traditionally have smaller rim/tire diameters, meaning you travel less distance per full turn of the wheel.

-In terms of tire design, most road tires are almost bald. This is because on a flat, dry surface you will generate enough friction to keep the tires from slipping. Mountain bikes have lots of knobs made to dig into the road/gravel/dirt, and tend to be a bit heavier. You may feel like you are less likely to slip with them, especially on wet road surface, but over a long ride they will make you less efficient and more tired. With hybrid tires, you're looking at the worst of both worlds. The center of the tire is mostly bald, while the edge of the tire is knobby. If you're trying to go for speed on a hybrid, like a road bike, you will find that the knobs on the edge fuck with the friction you're generating in a turn, and the tires are more likely to skid/slip.

If you're an every day rider going on short 10 mile rides you probably won't feel much of a difference, between a road vs MTB, but on a long ride, I promise you will feel it. I did my first ever century ride on an aluminum mountain bike where I swapped out the MTB tires for baldies to try and make it more like a road bike. My suspension system on the mountain bike could also be locked to make the fork stiff. My first ride took me 10 hours and my average speed was about 16 mph. The second year I switched to an aluminum/carbon fiber road bike and I cut my time in half, avg speed was 25 mph.

Is over training a thing? It sure is. If you aren't using cleats to clip into the pedals to take advantage of your opposite leg pulling up, while your main leg is pushing down, you end up having your quads doing all of the work, and they become overdeveloped. Overdeveloped or imbalanced quadriceps muscles can potentially contribute to a condition known as patella alta, or "high-riding patella". This condition occurs when the kneecap (patella) sits higher than normal in the groove of the thighbone (femur). While often a congenital or developmental condition, it can be influenced by muscle imbalances around the knee. As someone that experienced this first hand, it hurts, a lot. For me, this condition developed over 20 years of constant cycling; (I cycle to commute and exercise.) As long as you can do it safely, without inflaming old injuries, you should find ways to work your other leg muscles to keep things in balance to prevent this.

In terms of training for a century, everyone is different. I spent a year training on a mountain bike and gym bike/spinning before I even attempted a century ride. Before you sign up for a long ride sign up for shorter rides, sag supported rides. Start with 20, then 40, then 60 mile rides. (Sag support means there is a vehicle that follows behind the cyclists to offer support, or pit stops along the route to help with breakdowns, snacks, water ect.) Before I did my first century, on my bastardized mountain bike, I was able to do a 60 mile ride no problem. On the century ride, I hit my wall around 80 miles, the last 20 were pure torture.

Other considerations - learn to fix a flat tire, and make a point to carry spare inner tubes, a pump, and a pressure gauge, or pump with gauge for ANY ride that will take you away from where you can get reliable transportation or a ride to pick you up if you have a flat. In my area, the roads are absolute dogshit and full of major potholes, which are sometimes unavoidable resulting in a pinch flat. A pinch flat, also known as a snakebite, is a type of flat tire on a bicycle caused by the inner tube being pinched between the tire and the rim, typically due to underinflation or a hard impact. This results in two small, parallel holes in the tube, resembling a snakebite. Having your tires at the max allowable psi/barr will go a long way to prevent this, but you will experience more flats on a road vs mountain bike.

I am also an ex-runner. You will want a road bike for your uses. Unless it is a total piece of junk, the brand does not matter. So don't buy a bike from a department store, but any reputable brand (and even some that aren't very well-known) are just fine. The truth is, most of them are made in the same few factories anyway, then labeled with their brand. Fit is terribly, terribly important though. If the bike doesn't fit, you have nothing. And if you buy used, make sure you take it in to get a thorough tune up. If you have a bicycle collective where you live, they will restore older bikes, tune them, and sell them, which is ideal.

There might be an equivalent of C25K, but IMO, you don't need it. Once you get a bike that fits and functions, just go ride a lot. As one who has finished quite a few marathons and century rides, the century rides are honestly a lot easier.

Let me ask you this:

Are you looking for a bike purely for fitness, or do you have plans to get groceries with it, commute with it, go off-roading with it?

There's a massive range in bikes and what they can do... the joke that you always have "N + 1" bikes, isn't really a joke!

I've got a 90s steel mountain bike that is my hauler, bad weather, all-year-round, "it's only rain", "it's only mud", bike... but it's not practical for fitness/training.

I've got a folding bike that turns heads, but it's the slowest bike I've got, and I wouldn't dream of keeping it locked out front of a store for more than a second.

My last bike, however, is a gravel/road bike, and that thing is just for speed and distance. It's got no fenders, no racks, but can take me hundreds of KM at a time for fun.

I'd probably have 5 other bikes if space and money weren't a concern... so many bikes to do so many things!

Also, your goal to do a century in a few months as a runner, should be no problem! I went from ZERO riding experience only a few years ago to a metric century in under a month. A year later, 240+km in a single ride. This year, I hope to break 300km. Riding is fun! Enjoy, and yes, don't overtrain!

If you already use a Garmin device, the Garmin Cycling Coach (free version) works great, and can custom tailor your training to meet your goal, and will even adjust the schedule based on your progress and/or fitness levels.

I’m working up to my first century in early August. Prior to starting training in May I would do around 40 miles a week on a fixed gear. Most rides around 30-40 min. I am not in great shape and I have always enjoyed anaerobic sports more than aerobic.

I do a long slow ride on Sundays and then two one hour interval sessions on rollers during the week. With my starting long rides at 20 miles I increased roughly 5 miles a week up to 50 miles and then started increasing roughly 10 miles a week. I am very inconsistent at following this plan and this week I am doing a little over 75 this week. This is some shit training. My time is going to suck, but I am confident that I will finish the 100.

As far as overtraining goes, I’m not even very sore on Mondays and Tuesdays. Any misery I have felt so far in training has come from not eating and drinking enough. The symptoms there were akin to heatstroke and it took me a couple bouts to figure out what was going on.

The days where I start eating and drinking as soon as I get on my bike and repeat every 20 minutes have been my most comfortable and best paced rides.

As far as bike purchase goes, you are gonna be spending a lot of time on that bike. You want to be comfortable. I wouldn’t risk buying online. Go to a bike shop where they can help you buy something that fits. Not necessarily getting fitted though, that may blow a good chunk of your budget. If a shop can make a fit and bike work with your budget then definitely go for it. Whether you buy new or used, never buy an entry level model if you care about resale. The used market is flooded with entry level bikes and components. If you get into something higher level it will sell/ part out much more easily when you want to upgrade.

Edit: Washedupcynic had a really good point about clipless pedals/ toe cages/ straps. On long rides you want to be able to both push and pull. Don’t need anything fancy there though.