I just got back from apartment hunting in Fayetteville. There are a lot of apartments available for not much money. However, many, many places do not accept pets and some of the cheaper places are "tax credit" properties, meaning that you can't rent unless you are poor. Law students, or any full time student for that matter, don't qualify as "poor".
I don't have a list of furnished apartments but I remember that there are some available. I only looked at two apartments, the Razorback and the Cliffs II. The Razorback is about a mile from campus and is brand new. One bedrooms start at about $625 and come with a compact washer and dryer with a 13 month lease. You can find their website on the internet pretty easily. I thought that they were nice but they seemed smaller than I had expected there were not a lot of amenities, despite what they had on the internet. You could ride a bike to class very easily from there.
I ended up getting a two bedroom place at the Cliffs II. I think I ended up getting one of the last of two two-bedrooms available but there may be more opening up. There are also one-bedrooms. Two bedrooms start at $600 a month. The place is VERY VERY NICE. Make sure that you go by there even if you decide to rent elsewhere so you can get an idea of what is available for your money. I could not believe the amenities. Of course, what impressed me might not impress you (I liked the great weight room, the two-mile lighted running trail, and the fact that the community sits at the base of a small mountain that I can run my dog up and down every day). It's about a ten minute drive to campus. Just do yourself a favor and check the place out before signing a lease somewhere else.
While you are visiting I would recommend staying at the Radisson downtown. It is the biggest hotel downtown and is within walking distance of just about everything. They also have intenet access in a lot of the rooms (this was useful).