Software Update in Mountain Lion
Solution 1:
OS updates are in a different group in the Mac App Store, and neither checking for them nor downloading them requires you to be logged in to any account at all, so you should not have any problems.
Also note that you could simply enable automatic updates in System Preferences to avoid having to manage OS updates yourself.
Solution 2:
The term Software Update does appear in the menu, but clicking it opens the Updates tab in the App Store app.
