Check out these OWC portable external SSDs. Contact OWCs Customer Support to find out which of their portable SSDs will fit your MBA and needs best. In this case size or lack of size matters.
As for moving files to the external SSD the files that you have control over are located in the Documents, Downloads, Pictures, Music and Movies folders. You can use either of these two free apps, GrandPerspective or OmniDiscSweeper, to find the largest files on your drive so you can determine if they can be deleted or moved to an external HD for storage.
Note: you can empty the Downloads folder after the apps and/or updates that were downloaded have been installed or applied. Many users have found a couple of Gigabytes of files in their Downloads folder which are no longer needed.