How to get notified of new automatically archived emails in Gmail?
The steps you have to take are the following ones: 


Expand the list of recent labels, but this time tap the option Show all labels.

This will take you to the list of all your labels, in whose bottom you'll see the option Manage labels: tap there.

In the list of labels, tap the one you want to be notified about. 

By default that label has Label sync and notify disabled. Tap the option Sync messages
  
It will open a popup dialog with sync options. Choose and tap one different to Sync: None.

The above will enable the notifications, now you can choose its options according the way you want to be notified.

Exit the label options for changes to be applied. From now on, you'll receive notifications about unread messages for that label too (in Jelly Bean, you'll see the extended notification, as with inbox emails). You'll be able to open the email message tapping on that notification.