It is very tedious to remove only partial material, you have to search for every file you own on the machine and remove it if you change the admin user ID. You also have to be sure about transfer of the software you want to leave on the machine and licensing of that software, at the very least you must provide the new owner with all of the media for that software. Any apps you may have downloaded are permanently linked to your Apple ID and cannot be transfered.
If the MBA came with anything but Mountain Lion, as your info indicates it now has, you are required by license agreement with Apple to erase that Mac OS X version and reinstall the initial version. The new owner will not be able to update or reinstall ML if it was installed under your Apple ID.
By far the easiest and safest thing to do is erase the hard drive, reinstall the Mac OS X that came with the machine, and give the new owner the software medai that you have and let the new owner install what is wanted.