Wnen you connect to a computer with the Shutdown connection method, you will see three options:
I'm not sure what the difference is between "shut down" and "switch off", though I suspect that "switch off" is none too careful about closing programs and saving data, whereas "shut down" probably performs a standard shutdown sequence.
I rarely use this feature, because when you log in with full control, you can shutdown, log off, restart, etc. just as you would when sitting in front of the computer. So the answer to your second question is yes: you can install updates and restart just as you would normally.