Google recently updated their Gmail app for Android, which among other things adds support for Priority Inbox. It doesn't do much, though - you still need to jump through hoops to access it. Here's how to put it front and centre.
As of right now, Gmail only allows access to your Priority Inbox by hitting the Menu button on your phone, going to "Labels" and hitting "Important". It doesn't sort your email the way Gmail's web interface does. We've discussed adding shortcuts to your home screen before, but tech blog Tested gives you a perfect example of where it's useful.
To add a shortcut to your Priority Inbox, long press on an empty space on your home screen and choose Shortcuts. Scroll down to Gmail Label and choose Important. The shortcut will show up on your home scree, and you'll have quick access to your important messages in just one tap. Unfortunately, at the moment, apps like previously mentioned Gmail Label Notifier don't seem to recognise the Important label in Gmail. Thus, there doesn't seem to be a way to get notifications for only high priority messages just yet. If you figure one out, however, let us know in the comments and we'll be sure to update the post.