Mountain Lion: AirPlay doesn't show up (but it works in iTunes)
Airplay Mirroring in 10.8 is only compatible with machines that have a Core i-series processor. Your Core 2 machine is not on the supported hardware list for Airplay Mirroring with 10.8.

You should check out AirParrot, which is $9.99, but can do AirPlay Mirroring on older hardware (and OSes) as well.