It's not really that old - several computers dating back to 2001 have been reported to run Windows 7 just fine.
As Dave pointed out, Windows 7 is very good about getting its own drivers
for most hardware, and generally you'll just need to install videocard drivers and maybe sound and/or network drivers once Windows is up and running. But in this case, again, the Vista drivers provided by Asus stand a good chance of working fine, or you may actually get newer drivers from the video/sound/network device manufacturer's websites.
I also recommend downloading and running the Windows 7 Upgrade Advisor
(on your current XP of course) which will tell you about any specific issues you may encounter.
Finally, consider a dualboot with both XP and 7 installed; that way you always have XP as a fallback to boot into. (Plus, this will make migrating your user data and settings very easy.)
Planning a backup strategy, if you haven't already, is another prudent thing to do at this time - the free version of Macrium Reflect is an excellent drive imaging product to do this (it works in both XP and 7). EDIT
: there were a couple posts in the meantime (I was away for lunch lol). You'll be able to run Windows 7 fine with 1GB of RAM but you'll really benefit from an upgrade to 2GB.