Q: Can I use my nano to add missing songs to iTunes?

This explains all your options:

iTunes: Transferring media from your iPhone, iPad, iPod touch, or iPod - Apple Support

Basically, the iPod is not able to be used as a repository that can load your entire iTunes library.  Purchased items can transfer from the iPod.  Other items need to be transferred from your previous computer (or its backup).

Also see this:

How to transfer data from a PC to a Mac - Apple Support

Moving a library: Quick answer if you use iTunes' default preferences settings:  Copy the entire iTunes folder (and in doing so all its sub-folders and files) intact to the other drive.  Hold down the option (alt) key (shift on Windows) and open iTunes.  At the prompt to create or choose a library choose the copied iTunes folder. You would have to try really, really hard for this method not to work.

If this is to a new computer and you put the copied iTunes folder in the default location of Macintosh HD > Users > *User Name* > Music  then you don't even need to start with the option key held down, iTunes will automatically look for it there.  (Make sure there isn't anything already in the iTunes folder there that you want to keep since you will be replacing it with the one you are moving.)

Windows users see tip at: https://discussions.apple.com/message/18879381#18879381

Consider moving everything from your Windows PC to your Mac in a single move, including your iTunes library: About Windows Migration Assistant - http://support.apple.com/kb/ht4796 - Learn about how to use Windows Migration Assistant to migrate your files from a Windows PC to a Mac.

SemperEadem wrote:

 

What if I deleted iTunes off of my old PC, then reinstalled it and plugged my iPod in?

Doing this would not change the fact that your "iPod is not able to be used as a repository that can load your entire iTunes library" as I stated above.