Older Palm you had lying around? That thing's not old at all!! It will work just fine for Geocaching. And the "Palm Pilot" was a specific model that was introduced in 1997, and is ancient history. Don't worry though, they're never going to shake that Palm Pilot moniker.
Yes, Cachemate as suggested. Personally, I never tried it, I just go with Plucker. this is not Geocaching specific by any means, but allows you to view web pages on the Palm. Works great with Geocaching.com pages, or any webpage, for that matter.
I have a Tungsten E2, and I use Cachemate all the time.
I have an even older palm IIIxe (picked up off craigslist for $25) that I use cachmate ($8). GSAK exports a file directly for Cachemate. All you have to do is export and sync, works great! It is real nice having all of the information right in the PALM of your hand. You can download a sample of Cachemate and try it out.