Turn on automatic update
OS X has some support for automatically installing updates, but it's limited to "system data files and security updates" - what exactly constitute system data files, I'm not sure, but presumably any drivers that are typically distributed through Software Update, but possibly OS X point updates as well. Apple's documentation isn't clear on this.

You can enable this in the Software Update pane of System Preferences:



However this won't update any apps, and possibly not all OS X updates either (again, Apple's documentation isn't very clear on what counts as a "system data file").

The other solution you may be interested in is AppFresh, which claims to be a one-stop shop for all software updates, OS X or 3rd party. I haven't used it, but it seems to be able to integrate with and automatically install from Software Update (and they list Mountain Lion compatibility, so I'm assuming that includes updates through the App Store now), apps that use the fairly popular Sparkle updater, as well as Microsoft and Adobe updaters.