I think these are new: http://www.housesforrent.ws/details.asp ... 517707&s=r
It's about 6 blocks walk to the funky part of 12th south (great wine shop with free tastings every saturday, mafioza's restaurant, hip new cycle shop, one of the city's indie art galleries, improv theater). You'll pay more as you get closer to the street, and same goes for Hillsboro Village, but remember Nashville's small and you're no further than 2 miles from the law school so long as you stay within the I-610 loop. Actually, that apt is closer to 8th ave. s., which has Grimey's Records/The Basement (easily one of my favorite music venues, and not just because one of the co-owners grew up two towns over from me up in Mass). the neighborhood right by Belmont U is also pretty chill... houses on Belmont Ave are very nice. I would just look up apts and then streetview them on Google.
If you can find something between Belmont and 12 South that's north of, say, Kirkwood, you're looking at a 10 minute walk or less in either direction to both neighborhoods. It puts Hillsboro out at maybe a 15 minute walk and probably gives you a half-hr walk to the law school. I think you can definitely stay under budget... we pay 2,000 for a 3BR/2Bath house with a huge fenced in backyard in between Hillsboro and Belmont. I very much enjoy my Saturday afternoon strolls to the wine tastings and walking up the hill whenever my nights end up at the Villager in Hillsboro.
[edit: sorry, didn't see that one's unavailable.]