The first point to make is that an iPod is not a suitable backup for your music because there is no easy way to copy music on an iPod back to a computer or onto anywhere else.
A more suitable method is to backup the music that is on your computer to a suitably sized external drive. Buy yourself an external drive and copy the music to that instead.
Next, if you want to add music to the old iPod (but not as a backup), then if you still have iTunes, use it to copy music from your iTunes Library to the iPod. Alternatively, if you're using a newer Mac, use the built in Finder to add music to the iPod. If you are subscribed to Apple Music, that effectively makes it impossible to add your music to the iPod.
If you want to add Apple Music (the Apple Music subscription service) to your iPod, it will need to be a fairly recent iPod Touch. Apple Music cannot be added to other models of iPod.
Post back for further advice, based on which of the above scenarios you are using.