You have an Apple Store across the street, how convenient... I think you should just walk over and ask.
NOTE: Your iTunes library exists on your computer. Unless you subscribe to Apple's iTunes Match service (which matches or uploads your library's songs and playlists into iCloud), you cannot use another computer and access your library.
However, you can download previous purchases from the iTunes Store (at no cost) into a different iTunes library (not your library), and then set up the iPod to sync to that library, or set it to Manually manage music and videos, and manually load the songs (and other content) to your iPod. The manual method may be better in this case, since you won't be accessing that computer's iTunes library again.
To do this, go to the iTunes Store Home screen. Sign in using your Apple ID (the one you used to make the purchases). Click Purchased under QUICK LINKS. On the Purchased screen, you can find and download previous purchases. It may not be available for audiobook purchases.
After the iPod has the content, you should probably delete everything you downloaded in the computer's iTunes library.