Apple have published a page which clearly shows requirement specs for OS X Mountain Lion here:
http://www.apple.com/osx/specs/
Had you checked prior to purchase, you would not have been disappointed with Mountain Lion.
All I can suggest is that you reinstall your previous system then contact App Store Help and ask for a credit to your account. I would imagine a system profile would have to be submitted which would show you are not running the OS you're asking to be refunded.
I would point out to you, and please note, I am not employed by Apple in any way, that Mountain Lion offers significantly more improvements over previous versions of OS X, than just AirPlay Mirroring, like for example notification centre to name just one improvement.
I understand you may be slightly disappointed not having Mirroring, but on the bright side, you now have a good excuse to upgrade your older Mac.