Why does everyone think that once you get past sheetrock, air will travel outside somehow? Venting into a wall cavity will do NOTHING. There is a 2x4 along the floor in every wall, and (2) 2x4s along the ceiling. they're called plates, and your air will NOT travel past them.
Regarding your situation...i see you're on the first floor...does your Apt. have washer/dryer hookups? If so, you can tie into the dryer vent with a smaller inline fan. What about an exhaust fan above the stove? That's another possibility. A bathroom exhaust fan? <----careful here...sometimes, the simply exhaust into a cavity. Old chimney? Gas Furnace/Water Heater Flue?<-----Again, this can be tricky. Other than these, you'll be hard pressed to actually make it outside. If the exhaust air is filtered, why not exhaust near a window? If it's nice outside, the window can be opened. If it's cold out, keep the window closed and the extra heat from your room will help warm the rest of the apt. If it's hot outside then, chances are, an A/C vent exists just below that window and will temper that air before it makes too large an impact on climate feel.