Which linksys router do you have? Hopefully, it has a few switch ports.
First, disconnect the router from your neighbor's connection. Then connect your PC to your router on one of the switch ports and access the configuration page on
http://192.168.1.1. Go look at your wireless settings and make sure they are set the way you want it. Find the section on DHCP server and disable it. Since 192.168.1.1 is conflicting with your neighbor's, change it something else like 192.168.0.1. At this point, you will probably lose connection. Don't worry.
Now connect your neighbor's connection to one of the switch ports (NOT the WAN/Internet port). Make sure you get a light from that connection. I hope your linksys can detect that you are connecting to another switch (your neighbor's) and auto-crossover the connection. If not, perhaps one of the linksys switch ports is marked as being for connecting to another switch.
So at this point, your computer should be on your neighbor's network since you're both on the same switch. So if your neighbor is running dhcp, then your PC should soon get a new ip address from your neighbor and be able to access the internet. And if you correctly configured your wireless settings earlier, then you should be able to connect a wireless device and get an ip address from your neighbor's dhcp server.
As for accessing your router's config page again (
http://192.168.0.1), it might work without having to change your ip to 192.168.0.x.
Hope this helps. I *think* it works. :) Perhaps there is an easier way?