Is this wireless supposed to be a service for the whole building? If so, perhaps you should talk to the landlord. Airport isn't going to help if you don't already have a network connection in your apartment.
Do you have any friends who live there say closer to the lobby that get the WiFi with good signal? You can do what was suggested above if so. Otherwise I can't think of anything except the landlords do something like that.
My neighbor offered to share his wireless with me when we first moved into our home. The houses here are all brick and pretty large but I figured it was worth a shot. I decided to take him up on his offer. The signal was pretty spotty due to the distance between our houses and I would lose signal about every 10 minutes on my laptop when I worked in my home office. I bought two Linksys Wireless Bridges WET 11 that have large receiving antennas and look like little routers. I set one up on my home office and one upstairs for the family Imac. The WET11 could easily pull in the signal and we used them for almost 5 years. I started to do some video conferencing and needed my own high speed in the house but I saved 5 years worth of payments with this little investment in the Linksys bridges. You can find the WET11 cheap on Ebay. These are 802.11B but were plenty quick anyway. Give these a shot. If you really want to get clever, plug the WET11 into the area of your apartment with the best reception, then plug in an Apple Airport Express to transmit the WET11 signal throughout your apartment. I did this and it worked perfectly. This set up did NOT work with a WET11 and a Linksys router, but the Airport Express worked perfectly.
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