Like most cities in Arkansas, finding an apartment is possible throughout the year in Fayetteville. If you were to pinpoint one prime time to start apartment shopping though, it would probably be mid-summer. If you start looking mid summer, you shouldn't need more than about 6 weeks to find a high quality home.
If it is your first time getting your own apartment, don't be afraid to ask landlords if they will accept a consignor on the lease. This is someone who takes responsibility for making the payments should you find yourself unable to do so. Usually, this is a parent, but any stably employed person could take the role. If you've never rented an apartment before, this is a good way to get into an apartment and start building your rental history.
Fayetteville is small enough that it hasn't got a very highly developed public transportation system. However, this also means that everything is close enough together that a bicycle can be a great way to navigate the city. If you're studying or working at the University, Razorback transit provides free rides throughout the school year.
Watching a Razorbacks game should be one of the first things that newcomers try, in order to get a real sense of the enthusiasm that locals have for their community. The proximity to the Ozarks means there are opportunities for every type of outdoor entertainment, and Devil's Den State Park is a good choice for a first outing. The community is also home to one of only 3 Drive-ins left in the state, well worth a visit during the summer. Brewski's Restaurant & Draft Emporium prides itself on keeping the best beer selection in town stocked.