mac-appstoremacosnotificationssoftware-update
The App Store on Mountain Lion pushes update notifications through Notification Center. Unfortunately, this also happens for users that are not system administrators (and so cannot run the App Store). How can I turn this off?
Notification Center has preferences, where applications can be vetted, but App Store does not appear in this list.
When you purchase Mountain Lion in the App Store, it will first download an application called Install Mac OS X Mountain Lion.app to your /Applications directory. When the download is finished you're automatically prompted with the following window:
After that Mountain Lion installs and reboots into your Mountain Lion install. The installer application will be automatically removed - because most users won't need it anymore.
If you open the App Store, you're offered you to re-download Mountain Lion (hence the button says "download"), which could be useful if you wanted to create an install USB with the .dmg image.
You can verify if you have installed Mountain Lion by going to the menubar and click: Apple icon → About this Mac. In case you have Mountain Lion installed, you'll see this:
The notifications that you receive for Mail within Notification Center are local notifications, not push notifications from some Apple server. As such, the program has to be running in order for the action that will trigger a notification to be able to do so.
Notification Center cannot check for new mails by itself, it has no in-built capabilities for polling/checking etc in order to create notifications, it merely receives them and displays them.
It is possible that in due course some email app will be made available that supports push notifications, but at present none do, including Apple's own. The fact that you get such notifications on iOS is more a feature that the mail client is always running in order to check for new mail, and create the local notifications accordingly. Leaving mail open all the time on your Mac isn't really an issue, it's hardly resource intensive, and if upon receiving a mail alert you wish to act upon it, then opening the alert will only open mail for you anyway.
The best way to summarise it is that NC just listens for alerts that other applications create, and displays them according to your preferences. It takes no part in the creation of alerts.
Best Answer
You can now simply go to System Preferences > App Store, and turn off "Automatically check for updates."
No messing with firewalls, and just works with App Store. So just remember to check every now and then manually to find out when you have updates!