I am moving to an apartment that has WiFi throughout the building. I know that I can connect to the internet using the WiFi but will I be able to communicate wirelessly with my printer using my Apple Airport Extreme? I'm running Mac OS 10.5.4 on a G4 Powerbook.

Assuming that your printer has a wireless connection... the answer is yes. You should be able to create a network connection to the Internet (WiFi) and also have a device to device (or Ad Hoc) connection to your printer.