A while back I saw an apt. for rent for around $750 at the SW corner of Hudson & Fauntleroy (2 blocks south of Alaska). I haven’t walked that way in a few weeks so not sure if it’s still available.
Edited to add: I glanced over while passing on the bus this morning, and the sign is still there. There is also a house for rent on the same side of Fauntleroy between Dawson & Hudson, but I have no idea what the rent is on that – probably more than you’re looking at though.