We just moved from Manhattan to SoCo (Travis Heights) in April 2009 - that is south from Downtown zipcode 78704 and it is very young mainly, nice restaurants, about 20 min walk to dwontown, I assume the buses work too since we have seen those.
The downtown zipcode is 78701, I think Clarksville, Tarrytown are 78703 and probably just above (north) from downtown is 78705. We were told not to move to east for sure.
You can likely find lots of places for one bedroom apartment under $1,000
Camden | Home Page - good quality apartment complexes (many are) with pools, healt club (free), parking garage (free), business center (free)...contrary to NYC many places allow month by month rentals