You can sometimes find smaller houses in NoDa for rent around 800 or 900, but they are usually the dumpy ones that no one has touched yet.
I live in Villa Heights, one neighborhood down from NoDa (between NoDa and Racer's Belmont neighborhood.) We have friends in Belmont too and my wife was over there getting some eggs and lettuce from their house about 10 minutes before the drive by shooting at one of the sketchy neighborhood convenience stores about one block down the street. Personally my take on Belmont is that it would get better much more quickly if the city rezoned it and got those convenience stores out of the neighborhood.
Here in Villa Heights there have been a rash of break ins during the day lately. Well not a rash really, but 4-5 in the past month or so. There has been one stabbing in the year since we got here, along with occasional gunfire, but hey, "'Merica!"
Seriously though, if/when we buy a house I would love to buy in NoDa. We spend a ton of time there. (Hello Dog Bar.) But houses, and rent, is so much more affordable in Villa Heights, that we would look here, especially with the light rail extension. If we could find a place on that side of the neighborhood where you could walk to a light rail station, that would be ideal.
So that's a lot of rambling about our specific situation, but my point is you can probably find a one bedroom place, or a run down two bedroom place within your budget near NoDa if you really want to. But yeah, it would be better to know where your work is, etc.