If your budget is 400-500 you can definitely find something bigger and better although you will have to commute to school. Try just looking farther out of city center ( even Liberation could have a good situation with roommates for that price) maybe check leboncoin or nice accomodation facebook groups, I've found those to be helpful and student friendly.

I lived in Jean Medecin when I first moved to Nice and learned quickly it is one of the most expensive places in town rent wise and I could go a lot farther with my money other places in the city. I am not really aware of the student housing situation, but if you wanted to go on your own, there are definitely options.