Apartment cooling with fans
Ideally, you would blow warm air out of your apartment.  Cool outside air will naturally take its place.  If you can take advantage of prevailing winds by having open windows on opposite sides of the apartment, do so.  Otherwise, use a box fan blowing out of the screen door (assuming it's not in a bedroom -- they can be pretty noisy) and smaller fans pulling air in through other windows.

You can also keep the apartment cooler during the day by closing blinds.  If you have a balcony that gets sun during the day, consider getting some large plants: they'll provide some shade, and they'll need watering, which will also help to bring temperatures down.