I came across this utility called WUInstall: http://www.wuinstall.com/index.php/en
that completely controls every aspect of Windows Update installation. Has a free version and a Pro version. Free version has dozens of switches including check, download, install and reboot/shutdown automatically. Since Classic Start Menu allows for any custom command, we can add a command which calls WUInstall /silent /install /reboot or WUInstall /quiet /install /shutdown.
Edit: This is only required for Classic Shell versions prior to 4.0. With Classic Shell 4.0, the Start Menu now has integrated 'Install Updates and Restart/Shutdown feature'
I do the testing, some new feature ideas and UX/usability feedback for Classic Shell from 2009-present.