Well, Manhattan as a rule is quite expensive, especially if you don't know it super-well and you stick to the "well-beaten" touristy paths. So you might want to email med students from the school you're staying at (Sinai, I believe) and ask them about places to eat/ things to do around there. I'm assuming med students are even more po' than you are, so they should have good suggestions for cheap stuff in the area.
As for the car, it is absolutely not required and in fact I'd strongly recommend you don't bring one. Manhattan is a nightmare to drive in normally, and during that time it's even worse. And if you're not from there, chances are, you just have no idea how to drive there. I strongly suggest you look at the NY subway system map online and figure out where you're going to go and stuff. There are also buses and cabs just in case, but most places are totally reachable by subway. They tend to be a bit smelly and cramped, but oh well, that's part of the experience 😀
And again, Sinai students will probably be able to give you much better tips than I ever could.