I have approximately 90 GB of music and it is taking up too much of my Macbook Air's 256GB internal SSD space. I see I can buy a 128 GB flash card which can be left in the MacBook Air's flash card slot. Can I just move my library on to the flash card? Has anybody done this? Any advice? Would it be backed up with Time Machine correctly?
Not with a flash drive, but with an external drive. I've kept my iTunes library on a USB-attached drive for years now. Semantically, the situation is identical to storing it on the flash drive. The OS see's a remote disk in both cases.
Format it as Mac OS Extended (Journaled). Most flash media come pre-formatted as FAT32 these days. You don't want that.
For moving your iTunes library the correct way see: What is the safest way to move the iTunes Library from one hard drive to another?
As long as it's not listed in the Exclusions list in Time Machine, yes: it will be backed up. To check the exclusions list: System Preferences -> Time Machine. Click on the
You are able to specify what gets backed up with TimeMachine, so if you want to backup just your music you can do that and use the flashdrive as your TimeMachine backup. However, this won't allow you to erase the music off of your harddrive.
I would suggest simply putting all of your Music files onto an external (as you mentioned) by simply dragging and dropping the files, and then deleting it all off of your harddrive (except for any particular music you still want on your machine for regular access). If and when you want to pull a song off of your flash drive, just plug it in and pull it off.
Another idea is to look into itunes match (stores all of your songs on the cloud for $20 a year and you don't need to keep the actual file on your harddrive).