Is there any way to verify that OSX 10.8.1 installed correctly?
There is not really an easy way to verify the update went successfully. Your next steps should be to check your start up disk and then run a stand alone installer for the suspect OS X update. There is no harm in re-running an OS X installer over an OS with the same version.


Run Disk Utility on your start-up disk selecting Verify the Disk and also try Verify Permissions, selecting the repair options as needed.
Download the standalone OS X 10.8.1 installer.
Re-run the installer, and hopefully this time it completes without issue.