If you want a router, the Linksys WRT54G is probably the best. It acts as a router, an 802.11g wireless access point, and a 4 port switch (you can plug in four wired ethernet devices). It sits between your computer and your modem (assuming you have cable/DSL/some internet connection that plugs into your computer via wired ethernet), so it does not matter what OS your computer is running. I think the new ones have a one-press setup button, although for the most basic setup, you would just plug it in.
If you just want a USB wireless adapter, I do not know. You may want to check the prices and see what the difference is between buying a WRT54G and Nintendo's USB wireless adapter. You should be able to do it with any wireless card (PCI or USB), but I am not sure how easy it would be to set up on Windows ME. On 2k/XP I know you can just make a network bridge pretty easily.