Nope, I just installed the OS update on a machine that has not been linked to an apple id. The app store also has updates for the preinstalled versions of iphoto and imovie, but those do require an apple id.

I’m not sure if you can use the App Store upgrade channel without an Apple ID, but you can always download the update separately:

http://support.apple.com/kb/DL1571

It does not. OS updates are still delivered via Software Update, not through the App Store; ML simply integrates the SU GUI into the App Store app. For instance, I updated my 3 ML Macs to 10.8.1 this afternoon with the softwareupdate command-line tool, pulling the update from a local Software Update server, neither of which would have worked had the update only been available through the App Store.

Note that there's also a distinct 10.8.1 update to "Install OS X Mountain Lion.app", and that this is only distributed through the App Store — but this just slipstreams the Mountain Lion installer, and is neither necessary, nor even useful, for updating existing 10.8 systems to 10.8.1 (unless, of course, you choose to "update" by reinstalling the whole OS!).