Do you want the good news or the bad news?
The good news is that your price range puts you right in the sweet spot for apartments in Iowa City. You should be able to find what you're looking for in that price range.
The bad news is that the apartment market in Iowa City is TIGHT. It's going to be hard to find an apartment like that in the middle of the school year. You might have to compromise.
As far as neighborhoods, anything directly downtown is going to be older, likely hardwood floors, convenient and cool--but lots of crazy undergrads on the weekends. The Northside is going to be a good fit for you, anything on or around Church St, Fairchild St, Davenport St, North Johnson St.
You might also find what you're looking for on the West Side near UIHC or University Heights. Quieter, older, more med students and grad students. A little farther from campus, but much much quieter*. (*except for Football Saturdays)
Avoid S. Johnson St, S. Van Buren, S. Dodge, Bowery area. It's party central.
Here are some resources you might want to check out. If you're coming at semester, you might want to wait a while for some sublets to open up.
UI Off-Campus Housing
Service
Press-Citizen Classifieds
Daily Iowan Classifieds
Craigslist Iowa City