You seem to have gotten some shady information. Each iPod is happy syncing to a single computer (it doesn't care if its osx or windows). If you try to sync to a second computer (any operating system) it won't be happy. There are software programs for ~$20 that let you copy the music off your ipod without itunes (and therefore bypass the single computer restriction). Keep in mind if you have purchased music on your iPod, this may not be legal.
If you want to play music at work with the iPod connected, checking the 'manually manage" box (wording is similar to this, I don't have the right phrase), should allow you to play the music living on your ipod while it is connected.
Honestly the easiest solution is connecting speakers to your ipod (either with headphone or dock connector).