First make a deal with your neighbor and say he/she can play your Wii if he lets you play online using his internet connection. Or make a offer to him/her to secure their wireless router if they let you connect once in a while..or if all else fails offer to pay half the bill that always works =D
and to answer the connection problem the wireless routers ports are most likely misconfigred, to make sure all the ports are open for your Wii, just go to into the wireless router (http://192.168.1.1 most of the time) into the port fowarding settings area, and set the Wii's IP address as the DMZ or NAT connection.
Have fun!