To answer the question in your title, a wireless router can do anything that an wireless access point can do. An access point is basically a castrated router. If you want the most options now and in the future then go with a router.
In regard to your specific question, I would personally put a wireless router upstairs and connect it to your neighbors router using WDS. This will allow your router (upstairs) to pick up the wireless signal from your neighbors router and redistribute it in your house... so you can use your laptop anywhere. Then you need to connect your old printer (via an Ethernet cable) to the network as well, you can do this by putting another wireless router next to the printer and connecting that router (via wireless) to the router upstairs.
If the room your printer is in has wireless signal from your neighbors house then you could just get one router instead of two, and that router would connect to the printer via an Ethernet cable and would also connect to your neighbors wireless via WDS. If you get a good router with big antennas this may be possible.
Another option is simply moving your printer upstairs where you know you have a good signal :)
I would make a final suggestion, make sure the router you get supports DD-WRT firmware. DD-WRT lets you do a lot of stuff and has an excellent interface. DD-WRT will allow you to boost the wireless transmission power of your router making it easier to connect to your neighbors network, a feature that is not available on any OEM router that I am aware of. You can always go with the good ole stand by, the Linksys WRT-54G, a router that has stood the test of time. I have owned over a dozen of those routers and they perform very well. Lately though my router of choice has been either the Netgear DIR-815 (A/B/G/N networks, two radios) or the Netgear DIR-615 (B/G/N networks, one radio).
