Houston is huge. I've been here six years and there are plenty of areas I've never been to (some I'd like to explore, some I have no desire to see). Each area has its own character, upsides, and downsides- there's not so much a sense of the city of Houston overall, which can make it feel a little smaller than it really is. Learn the area immediately around you first, then branch out slowly. This goes triply if you don't bring a car, since you really can't get very far.
You can live alone on that stipend within biking distance of UH, but not particularly safely. You'll want a roommate to split costs. I doubt you could afford to keep a car and live on your own. The area around UH is not very nice, so unless you're okay with bumming rides whenever you want to go somewhere interesting, a car is pretty useful to have.
One of the (few) upsides to Houston is that, since it's flat and doesn't get very cold, it's a very bikeable city. I can't say it's a really convenient bike ride from UH to Montrose/Midtown/other fun areas, and I wouldn't want to commute from Montrose to UH by bike every day, but if you live relatively close to campus it's a good way to get out.
