The easiest way to do this is with iTunes.
Turn on Home Sharing for your iTunes account and enable the feature in iTunes on your NAS-connected machine and on your machine you wish to use as a source for material to copy.
Start iTunes on both machines.
From the MacBook Air machine, look for your other iTunes library under the SHARED section in the left-hand list of iTunes. Select your other machine's library and expand the library contents. Select 'Music' to show all the music in that library:
Select all the music in that library by clicking on the top song in the list and pressing Cmd-A to select everything:
Now click the 'Import' button in the bottom right corner:
iTunes will copy all the music from your large library in to your MacBook Air's library, storing the data on your NAS. You can speed the process along by doing the copy over a wired network connection instead of WiFi.
You can repeat this process for all content in the remote library, not just music.