2011 MacAir will not run Thunderbolt
Great! I've had my TB Display for almost a year (and, yes, I purchased the AppleCare Protection Plan for it, so I've another two years remaining under warranty). I love it - only disconnect from it every two weeks to condition my battery.
If I can save my pennies, in 2015 (when the warranty runs out on my MBP and Display) I'll buy a Mac Pro and an Eizo monitor... until then, what I have works well for my needs.
Enjoy your new (working) monitor!