Gizmodo shows us that you can get it done using Google's free VOIP service, Google Voice. Once you sign up for Google Voice, you'll be assigned a unique phone number in whatever area code you choose, so it'll match the local area code. Give that one to your landlord to program in the building's call box, and make sure Google Voice's settings are set to forward that call to your main number (probably your cell, right?).
From here, you'll need to mess around a bit with Google Voice's settings. Once you buzz yourself from your building, you'll be able to save the apartment's call box as a new contact and, using Google Voice, assign that contact to a group that can ring multiple phones at once—all you need to to is add each of those lines for forwarding in Google Voice.