Make sure that your Windows firewall is configured to allow LW.
Firewall setup

Try using this version of Java. Use the JRE offline installer & remove your current Java version before you install the new one.
Archive: Download Java 2 Platform Standard Edition (J2SE) 5.0 Update 11

Your router has a firewall. Try port forwarding. The instructions are here.
Port Forwarding for the Linksys WRT54G
Don't forget the step of setting up a static IP address first LW's listening port, port in the 'manual port forward' box & port forwarded within the router must all be the same number!

If you still have problems, maybe there's a firewall associated with the apartment complex's internet? You'd have to ask the complex's manager about that

I'd advise you to get some AV protection if you're going to be using LW...there are heaps of infected files being shared! Check the link below for info about reliable, free AV etc.
http://www.gnutellaforums.com/264170-post2.html