Q: Syncing ipod Nano to alternative computer

If you are using automatic syncing, your iPod can only be associated with ONE iTunes library at a time.  It's not the "computer" or the computer being "authorized."  By definition, iTunes can only "synchronize" your iPod with one iTunes library.  If you have a backup of your complete iTunes library, you could restore your iTunes library to a new computer, and your iPod would happily sync with it, because it's the same iTunes library (on a different computer).

When you associated your iPod to a new iTunes library, its existing content from the old iTunes library is erased and replaced with content from the new iTunes library.  To avoid losing the iPod's existing content, you need to off-load it.  By design, iTunes syncing is typically in one direction only, from iTunes library to iPod.  You cannot use iTunes to transfer songs from iPod to computer (except for songs purchased from the iTunes Store).  However, there are third-party methods and utilities that can transfer from iPod to computer.  If you do an Internet search on something like "ipod music transfer," you should get some useful links.  Here's one method that does not require any additional software

http://www.cnet.com/how-to/how-to-transfer-music-from-an-ipod-to-your-computer/

Once the iPod's song files are on your computer's drive, add them to your current iTunes library.  After confirming all songs are in your iTunes library, you can sync your iPod.  Going forward, you should back up your iTunes data, along with the rest of your user data.  iPods are not designed to be your iTunes backup.  To back up just your complete iTunes library, you can use this method

Manage and back up your iTunes media library - Apple Support