Q: Is my MacBook Pro too old for AirPlay?

AirPlay Mirroring needs an Early 2011 or newer MacBook Pro and Mountain Lion > http://support.apple.com/kb/HT5404

As your computer isn't supported, you can use an app like AirParrot > http://www.airparrot.com

About AirPlay and Airplay Mirroring

AirPlay Mirroring requires a second-generation Apple TV or later, and is supported on the following Mac models: iMac (Mid 2011 or newer), Mac mini (Mid 2011 or newer), MacBook Air (Mid 2011 or newer), and MacBook Pro (Early 2011 or newer). For non-qualifying Macs you can try using Air Parrot.

Several Apple Articles Regarding AirPlay

Apple TV (2nd and 3rd gen)- How to use AirPlay Mirroring

How to set up and configure AirPort Express for AirPlay and iTunes

About AirPlay Mirroring in OS X Mountain Lion

iTunes 10- About playing music with AirPlay

Troubleshooting AirPlay and AirPlay Mirroring

Using AirPlay

Thanks to the $15 Beamer, AirPlay streaming is still possible on Macs  that do not support Airplay and mirroring.

Another solution is the Air Parrot.

You have AirPlay to transfer iTunes data to your Apple TV, but you have AirPlay Mirroring to mirror your screen. AirParrot only allows you to mirror the screen

A better solution for using your TV as a monitor is to connect an HDMI cable from the TV to an adaptor that in turn plugs into your computer's display port. In this way you get maximum video speed and the TV functions as a true second monitor.