Unfortunately, there is no API provided by MS for "Install Updates and Shutdown" functionality. But you can customize the Shutdown actions on the Customize Start Menu tab.
Open Classic Start Menu Settings, enable All Settings radio button and go to Customize Start Menu tab. In the left column, double click "ShutdownBoxItem". Its "Command" is the default action. You can set the default to whatever you want.
You can also add a custom command or item for "Install Updates and Restart or Shutdown" by dragging a "Custom Command" from the right column. There is this free third party tool called WUInstall:
http://www.wuinstall.com/index.php/en that completely controls every aspect of Windows Update installation. 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 to WUInstall.exe which calls the right switches. e.g. WUInstall /silent /install /reboot or WUInstall /quiet /install /shutdown.