I'm not sure if there's any way to do it easily on google apps, but on my main e-mail host (uses merak mailserver), i've set my noreply address to use an autoresponder at all times, and also set it to automatically delete e-mail after 1 day.
I'm unsure about automatically deleting email on google apps, but you can set up an autoresponder and leave it turned on all the time.
Then the only reason you would ever log into it is either to send e-mail, or to erase emails from the inbox (if I log into mine and notice emails in it, I never even bother reading them....I just completely ignore them, and my hosts mailserver deletes them automatically).
