Riding mountain bike to work regularly?

It's hard to get a better bike than a mountain bike for going up and down hills. Even a cheap one. The geometry is set up so it's controllable both up and down steep hills, and the gearing almost certainly goes low enough that you'll be able to pedal up even the steepest of hills (as long as you're strong enough to push the combined weight if you, your bike, and all you're carrying up that hill...)

For going down steep hills, as long as you're not trying to set any speed records you just have to be able to stay in control. (I'm ignoring brakes overheating on long descents here...)

For going up steep hills, as long as the bike is geared so you can pedal it up the hill fast enough to stay upright, the only thing that really matters is the combined weight of the rider, the bike, and everything else you're carrying. Shaving weight off even the cheapest and heaviest of bicycles won't make a huge difference here, and even the cheapest, heaviest mountain bike probably has low enough gearing. Shaving just 5 lbs or 2 kg from a bike can get expensive. The easiest place to lose any weight is to not carry anything heavy. The next easiest place to lose weight is you...

One thing that can help is using either toe clips or clipless pedals to allow you to also pull up on the pedals. That will allow you to engage more muscles as you climb and that could help take some load off your quads and glutes. Note, though, you'll still have the same cardiovascular limits on how much total power your muscles can produce.

Finally, though, it's all about you and your fitness and strength. Those will both improve as you ride more. The ride will get easier and/or you'll go faster.

This is a common conundrum for new bike commuters! Yes, there are certainly bikes out there that will be better for you at hill-climbing or commuting in general. However, it's hard to know if it's worth the investment until you know for sure that you want to continue commuting by bicycle... but it's hard to know if you want to continue commuting by bicycle until you've tried it for a few weeks. A classic chicken-and-egg problem!

My suggestion is to take a few weeks to get used to riding the bike up and down those inclines before you try it out as a commute. If you tackle the hill regularly, your body will start to adjust and it will get easier, all other things being equal. It's a question of leg muscle, sure, but it's also cardiovascular efficiency, core strength, etc. With practice, your body will realize, "oh, I guess I'm going to be asked to do this every day now - I'd better prioritize these systems!" and the ride will start to get easier.

One of the most important factors in your ability to get up the hill on the bike you currently have is bike fit. The answers to this question will give you a lot of detail about what factors make a bike fit you properly. Adjusting the saddle and handlebar height will get you a long way toward better performance up the hill and better comfort on the bike in general.

That will give you a boost out of the chicken-and-egg cycle. Once you're more comfortable on the bike and more able to tackle the hill in question, that will open up the whole world of bike-buying at your feet! If it is something you want to do long-term, I would think about some kind of commuter or hybrid that favors the "road" style (more so than the "mountain" style). Test ride lots of different bikes to get a feel for what style you like, what sizes fit you comfortably, etc. This depends on where you live and what's available, but used bikes can be a great investment if you have a trustworthy shop or mechanic to buy them from. You can get a lot "more bike" for the same amount of money with a good used model. A well-maintained bike should last you quite a long time no matter when it was made.