Welcome to the Apple Support Communities
To have the App Store, you need Mac OS X Snow Leopard, and you can't get it from Software Update because it's a paid upgrade. Buy it from the Apple Online Store > http://store.apple.com/us/product/MC573/mac-os-x-106-snow-leopard
Then, make a backup of your files with Time Machine and/or Carbon Copy Cloner, insert the DVD and follow the steps to upgrade to Snow Leopard. After upgrading, open > Software Update to install 10.6.8, so you will get the App Store.
To upgrade to OS X Mountain Lion, just open the App Store and purchase it. Check that your applications are compatible > http://www.roaringapps.com When it finishes downloading, just follow the steps to upgrade to Mountain Lion. You can check the OS X Mountain Lion requirements > http://www.apple.com/osx/specs