How did you install Mt. Lion? On top of your old system or did you make a clean install? What was your previous system?
Where have you checked for Aperture? It should be installed in your Applications folder. Have you searched with Spotlight? If not, type "Aperture" into the Spotlight searchfield and see, if any Aperture applications are found.
If you see it in the drop-down menu, select it and hold down the ⌥⌘-key combination and double-click it to reveal it in the Finder. And select it in the folder that opens and drag it back to the Dock.
The previous version was an upgrade to Aperture 3 via download so no discs exist. What next?
Did you buy an upgrade key and used it to unlock the Trial version?
Then you can reinstall Aperture from the Trial version and unlock it with your serial Number: see http://support.apple.com/kb/TS4412?viewlocale=en_US
If you originally installed Aperture 3 from a trial version:
- Go to the Applications folder and delete Aperture.
- Download the Aperture 3.1 Trial.
- Open the Aperture Trial disc image.
- Double-click the "Aperture Trial.mpkg" file.
- Follow the onscreen instructions to install Aperture.
- After the installation completes, choose Software Update from the Apple () menu.
- Install the Aperture 3.4 update from the list of available updates.
Note: If you are using OS X Mountain Lion v10.8.2 or later, you may receive an alert stating that Aperture is from an unidentified developer. This can occur when you try to install Aperture from DVD or the trial download. See OS X Mountain Lion: The app is from an unidentified developer for details on how to successfully install Aperture in this situation.