How much cheaper is Yokohama compared to Tokyo? Can I live there with a budget of around 120,000 yen a month?
It depends, but you could do - barely. I'm quite lucky in that, as a student of Yokohama national university, my rent is heavily subsidised - I only pay 14,800 yen a month, plus another 5,000 or so in electricity bills. Normally you'd be looking at a good 40-50,000 in rent, plus electricity. You also need to take into account food etc, and transport costs (depending where you live, what lines you might need to use regularly, and how often you need to use them). I'm also very lucky in that getting to uni only requires me to use the yokohama subway line, and because I commute there daily, it made sense to get a commuter pass. And because nearly everywhere else I'd ever want to go to in Yokohama is somewhere between me and uni (it's ten stops away), I can go basically anywhere I want at no cost using my pass.
If you take all that into account, I'd say 120,000 yen a month is very doable. You wouldn't have much money left over to enjoy yourself, or spend on anything beyond living costs though.
I see. How much do cars cost there? I looked up the licensing stuff and it wasn't bad if you're not going to hire a translator. I don't think I can do the train stuff.
How does Chiba and Kyoto fair against yokohoma?
No idea. But trying to drive a car around in any of the big cities - Tokyo, Yokohama, Osaka, etc - is not going to work. I know you're American, so you probably don't really 'get' public transport in the same way people from Europe or Asia do, but going by train is far, far more efficient (and definitely cheaper). It is without a doubt the best way to get around here, and actually kind of fun (provided you avoid rush hour, in which case it can be a bit difficult

). A lot of trains are probably more comfortable than any car, as well (if you're sitting down, of course).
The public transport system does take a little bit of getting used to - it took me a couple of attempts to work out the difference between local, express, and limited express trains - but it's not that hard really, and worth the effort. Most signs are in English as well as Japanese too, which really helps.
I've never been to Chiba (there's nothing there, really, so no reason to). I've never been to Kyoto either, but I'll be going there in a few days for a week, so I'll be able to give you a full report fairly soon

^^ I set that budget based on the ammount MEXT scholarship is offering. Assuming I can get it, I guess I can study in Japan without having to starve everyday.
Ah, I think I know the one you mean. You should be able to manage it, and Yokohama's no more expensive than any of the other cities I've visited so far (Sapporo, Kumamoto and Fukuoka), so it should be fine. I have a couple of friends living off an 80,000yen/month scholarship, and they seem to be managing (but again, this is with the reduced rent), though it's apparently quite tight. 120,000 should be perfectly doable, though saving up a bit of your own money to complement it will really help you make the best use of your time here.