Well, my question is do I need both a wireless router and wireless print server, or just a wireless print server.
Yes you need both. You need a router to make a proper network. Your router is the hub that organizes all PCs and devices on your network. it also has tools for your first line of defense when adding internet to it. A router is a critical piece to doing anything you want for home networking. The print server really does nothing more than add a wifi adapter card to a printer.
However, I foresee another problem for you (and you neighbors).
You, I assume, want to keep leaching the weak signal you get from them for your internet access while accessing your printer. This isn't possible in an easy way. Here are some solutions.
1. Buy just a print server and see if their wifi signal is strong enough for the print server to connect to their wifi network. This will give them access to your printer and you access to their internet.
I personally don't like that. For one I don't like accessing someone else's wifi router. They are now in control of my central hub of networking. It would have to be someone I really trust. Even though I could run a firewall and monitor all activity, you still are making it easier for someone else grabbing that same signal to begin attacks on the security of your PC. Since their wireless is unsecured, this means anyone driving through or in the neighborhood grabbing that signal has the first door open for them to attempting access to anyone else connected to it.
2. Buy your own wireless router and a print server. Use Virtual wifi to connect to 2 different wifi connection. I still don't like the unsecured nature of being on someone else's network. Virtual wifi allows you to connect to multiple wireless networks by using switching. So you can make a wireless network at home where only you have access to devices on it and you can still connect to the neighbors internet.
3. Buy your own wireless router and a print server. Then just simply connect and disconnect to which network you want at the time. A bit inconvenient, but honestly you don't really need the internet and your printer at the same time to often. Even if you need to print agreements from stuff you order or do online, you can save these agreements locally to your PC, then disconnect from the internet wifi and connect to your printer wifi and print from you PC.
I'd personally do none of them and forget about connecting to the neighbors and pay for my own internet. //content.invisioncic.com/y282845/emoticons/laugh.gif.48439b2acf2cfca21620f01e7f77d1e4.gif