I don't think that message is related to the iPod being synced to another computer... It usually means some other application or process on the computer (that is not iTunes) is accessing the iPod for some reason, so iTunes is not able to perform the update. However, your question in the topic subject ("How can I use an ipod nano that was synced to another computer?") is valid, since the iPod was previously used by someone else, with a different iTunes library.
If you want to start using this iPod with your iTunes library and put your songs on it, instead of doing an Update, you should to a Restore. This erases the iPod, reinstalls the latest software, and sets it to default settings. The iPod gets a "fresh start," like when it was new. To resolve the issue with this error message about "files in use by another application," try restarting your computer. Immediately after the computer finishes starting up, run iTunes, connect iPod (if it's not already), and try doing the Restore (or Update). If you still get that error message, please post back.
If you do the Restore, you can set it up to sync with your iTunes library. You can post back, if you need advice on the options for setting it up.