Can I get push notifications for emails in my Gmail Priority Inbox only?
After days of searching I have found the solution. https://www.ultimatenotifier.com/


You have to buy Ultimate Notifier from the AppStore. $2.99 at the moment. 
Create a free account on their site or through the app.
Login to the web interface on your computer and under Add Notification select Add Gmail / Google Mail - NEW Priority Inbox
Complete the fields username and password with your Gmail username and password. (I know this is a BIG leap of faith.) 
Save.
Log in to the app on your iPhone to receive push notifications