The usual setup is
INet <-> Wireless router
That is, both system should be able to connect directly and independently to the router, and the router will dynamically assign addresses and DNS settings from its pool of available addresses.
For example, my son has a Qwest router in his house with two desktop PCs and a laptop PC (all running XP). One desktop is hardwired to the router, whilst the other desktop and the laptop use a wireless connection. Each wireless connection is automatically made when the system is powered on.
When I'm at his house, and turn on my Linux laptop, I'm also automatically connected to his network.
(Note: His home network uses WPA2 pass-phrase encryption, so the "automatic" connection part only happens because he and I have "saved" the pass-phrase for the various connection software systems to use when the connection is being established. In the XP system this is "automatically" done when the phrase is first entered. On the Linux system, it's stored in a "wallet" that needs to be opened before the connection software can access it.)
If you really
want to use the XP a secondary router, you'll need to purchase equipment (a wireless router with NAT capabilities, an Ethernet card, cables, etc.) to set it up. (This is possible to do, but expansive and, for what you describe, probably pointless.)
One "wrinkle" to all this is that, by default, almost all wireless routers are set up to assign addresses from the 192.168.-.- pool. Any device with an address in that pool can only communicate with the router, but not with any other device connected to the router. In other words, if your neighbor's router uses the default configuration settings, your two systems will be connect to the Internet, but they will not be able to "talk" to each other.