[Marvel 33]

I'm not sure if you could find an $800 apartment in Lincoln Park or even in Lakeview. Maybe for a vintage gardern studio-apartment in an older building but not a one bedroom.
In that price range I would probably go with Ravenswood, Edgewater, Rogers Park, Ukranian Village, Logan Square, Lincoln Square, some parts of Wicker Park and Roscoe Village. Of these neirbohoods my first choice would probably be Lincoln Square - wonderful neighborhood, active night life, a great variety of restaurants, safe, close to public transportation (brown line).
If you could spend little bit more ($1,000 - $1,200) then I would look in Lakeview, Lincoln Park, Old Town or Bucktown.
Good luck!