I just needed to do this and it was fairly easy using this method:
Install IFTTT (free app that supports all kinds of awesome rules).
Create a new "applet" where:
IF This: checks gmail for messages according to whatever criteria you want
Then That: send notification
I'm doing this to try to get exchange messages from work (but cannot access or send content outside of work). I wound up creating a rule in outlook that sends a new message to my gmail account with a set subject/body (via a script). IFTTT searches for new messages with that subject and pops up a notification.
I used this page to help me create the script to create a new message:
I saw that you don't trust Google with your personal mail. The gmail account you use could be a dummy account (I assume you may already do this since you're on android). Or you could use an alternative trigger from IFTTT.