To upgrade past Snow Leopard. For Mountain Lion, the minimum specs are ... technical specification
Start by running software update to make sure your 10.6.8 is the latest.
You can then go directly from Snow Leopard to Mountain Lion via the App Store, providing you have met the specs.
Double check the memory requirements because Mountain Lion tends to be memory hungry. At least 4GB and 6GB if possible.
If you have the original, 1st edition, MacBook Air, you'll be limited to Lion. In which case you would have to phone Apple, 1-800-My-Apple, to get authorization to purchase Lion, as it is no longer shown in the App Store.
If you have one of the newer MacBook Air's, you can run Mountain Lion.
Either would have to be purchased through the App Store.
MacBook Air's RAM can't be upgraded. You have what you originally ordered.
P.S., my wife has an original MacBook Air 1.1, it has 2 Gigs of RAM, and is currently running Lion just fine.