Hotel Bologna and Hotel Giuletta e Romeo are in that area, but might be slightly above your budget. Try looking at www.hoteltorcolo.it, which isn’t that far from Piazza Bra. As a reference point, its about 20 minutes walking from the main train station to Piazza Bra.

I really like hotel Bra on the piazza, its wonderful. The breakfasts are good, it’s not noisy and it’s pretty central to everything. The only thing is it’s a bit far from the station so I’d suggest hiring a taxi as they are not too expensive.