Sometimes you can find cars for around 500-1000 that run just fine but look terrible. The problem with going with a really cheap buy is if you do take a long road trip something is very likely to happen and then you could be stranded in the middle of nowhere. Renting to drive that many miles is going to run you quite a bit of money but will be a lot safer.
Maybe you could think about doing both, buy a very cheap perhaps even late '80s 500 dollar beater car to use for the year and then rent for the road trip. Truly your best option would be to find someone to take the trip with you, if you are studying there I am sure you can find someone at your university that wants to road trip as well, they may even have their own car.