Notification of new label e-mails
Don't know quite how you've got it configured to show your non-inbox mail alongside your main mail in the web interface, but if you're using the Priority Inbox features of GMail then you can set up separate alert settings on Android for mails detected as priority messages, see this Google Mobile blog entry: Gmail for Android: better Priority Inbox support and improved compose


  Ever wanted to know that you got an
  important message without taking your
  phone out of your pocket? Now you can
  set up your phone to notify, vibrate,
  or ring on just your new important
  mail (check out Menu > Settings >
  Priority Inbox).