I think it's doable for you to find rentals in that area for that price, you just may have to hunt around a bit. I used to have a friend who rents a duplex there with a small yard for about $950. The areas that are popular with young folks, like Alameda, all have high "walkability" scores, near cafes, restaurants, shopping, etc. Other areas you might consider include Hollywood, Hawthorne, Concordia, Kenton, Mississippi and the area around Mall 205 (for example Cherry Park). There is a website you can go to where you can find that out the "walkscore" for virtually any address in any city in the US: http://www.walkscore.com You can also go on http://www.rentometer.com to see what the rents are in a given area... start out by putting in address of something you know is for rent, even if it's over your price, and go from there. I think rentometer.com is listing rentals now too, so you might even find something there!
Anyway, hope this helps, and good luck!
Realtor & Landlord