I grew up in Westwood so I can attempt to give you some insight. Most UCLA students I know live within walking distance of campus (granted, UCLA is in a very pricey part of town). Public transportation isn't the greatest, but if you get housing on one of the Big Blue Bus lines, you'll be fine (the 1, 2, 12, etc). There are also some UCLA shuttles that I believe pick up and drop off around west LA, but I'm not too sure about the specifics on that.
As for apartments with cars...this is LA so you are pretty much expected to have a car, and therefore most apartments have parking (in fact, I think now all new apartment buildings are required to have subterranean parking).
I think a 1br within walking distance would put you around 1200ish.