If you can get a SSD installed in your machine, you will see a HUGE improvement in performance. The Intel Core 2 Duo is a very good chip, your RAM is good, except you only have 2gb which is a bit low.
On my Late 2010 MacBook Air, with a SSD and a Intel Core 2 Duo, and 2GB of Ram I am running Mavericks and it runs amazingly well, and very cool except when doing something very intense, which is where SMCFanControl comes into play. I crank the fan up to the highest it can go and that prevents the system from getting so hot it cannot perform well.
OWC has SSD options, though I am not 100% sure if you can get one because you have HDD. I believe it is possible, here is a link:
http://www.macsales.com (It's OWC's Mac Store)
Here is the specific link for your machine and SSD's, which when you look at prices, it is MUCH cheaper than buying a new Mac! :
http://eshop.macsales.com/shop/SSD/OWC/Air-Retina/Apple-MacBook-Air-2009-Drive-Internal
Here is a video from OWC, showing you the difference a SSD can make vs. a traditional HDD:
http://eshop.macsales.com/owcpages/ssd-bootest-electra-110113.html
It's either you do that, or buy a new a Mac. However, I sincerely believe if you install a SSD your system will undoubtedly perform extraordinarily better, and cooler.
My opinion is you get a SSD installed from OWC (if you have the money) which is much cheaper than buying a new system. Also, when you send your Air out to OWC for the install they will clean the interior if your system, like your fan which as I said in a previous post that could be clogged a bit. It is very important to keep the fan, and the vents very clean for maximum cooling. Also, a cheap option to help is to use a cooling board, which you can buy at any Best Buy etc...
If you have anymore questions, just ask me and I will respond as soon as I can.
Hope this helps!
Edit: Snow Leopard is the best OS X made, but as someone else said you should be able to run Mountain Lion decently well. Also, get some canned compressed air, and spray it around your cooling vents, you might be able to get some dirt out of the way. The vents are hard to see but they are there.
Here are some links to articles about cooling on your Mac:
http://apple.stackexchange.com/questions/7319/what-areas-of-the-macbook-air-should-i-keep-clear
http://support.apple.com/kb/ht4543
https://forums.macrumors.com/threads/1310635/
http://t-machine.org/index.php/2010/12/25/how-to-cool-down-a-macbook-air-fast/comment-page-1/
Also, NEVER try opening your Mac unless you truly know what you are doing, and even then it is risky to do so. Always have a professional that is trained to do this kind of thing perform any thing like this.