You do realize the University of Arkansas is nowhere near Little Rock, don't you?
There is a branch there, actually, and the library website shows it has some fairly interesting archival material and special collections. Who knew?
UALR and the Arkansas State Archive are in Little Rock. However, the main University of Arkansas campus is in Fayetteville, and the library in Fayetteville also has some archival state records. Best to clarify which campus before making plans.
That said, the neighborhood near UALR and the Arkansas State Capital complex is generally pretty safe. I'd suggest contacting the UALR library, the State Archives and/or the UALR visitors' center for lodging recommendations.
+1 to all this, including clarifying which library. Also the Clinton library is in Little Rock, which might be useful if what you are doing is political history.
My impression is that mass transit in Little Rock, while it exists, is nothing special (here is the website: http://www.cat.org/
However, if you stay close to the UALR campus and the archives, maybe you could get away with only renting a car on occasion. On that note, check to see if UALR has any visiting scholar facilities, or maybe the opportunity to stay in a dorm. The regular semester will be over, but summer school should be in session, so it might be a possibility.
While you are there, take the opportunity to see some of the sites, both scenic and otherwise. Petit Jean State Park (on a mountain and the site of Winthrop Rockefeller's model farm and antique car collection), the Ozark mountains (which are really, really beautiful) and Hot Springs, the home away from home for 30's gangsters, are all about an hour away in various directions. Fayetteville is about three hours away, but a really lovely drive (you can go the long way, interstate all the way, or take I-40 to 71 and a curvy drive to Fayetteville, or take the Pig Trail (Highway 7, I believe) almost all the way from Little Rock (it is very curvy but very beautiful). Eureka Springs is also in the same vicinity as Fayetteville, but closer to Little Rock. Here is one website for attractions: http://www.arkansas.com/things-to-do/mustsee/
There are some really kitschy and crappy tourist attractions around (as throughout that area of the south) but also some very interesting and beautiful areas.