Yeah, you should just be able to plug in the iPod to the computer your music is on. It should work like normal. Shouldn't be a problem. :)
bump for later
If the iTunes application isn't allowing a second iPod to be programmed to the computer you're using, 1) call Apple, and if that doesn't work 2) return the iPod. (I would guess, however, that Apple allows more than one iPod to register with a single PC -- bad marketing decision if they don't.)
One other thing: It might be that your PC isn't reading your iPod -- this happened to me when I tried to program my Mini. The problem wasn't with the software or the computer, but with the fire-wire connection between the computer and the iPod. My son, who worked at CompUSA, said that faulty fire-wires were the #1 complaint from new iPod owners. Use the fire-wire from your other iPod, and see if that works.