Phone ring for email
It is possible through the use of an app called IF. It was previously called IFTTT (If this then that). It is a great app for things like this. It has many uses including the one you have described and for many other things as well. You could have it send you and SMS when a certain event happens, if you have the HUE light system it could change the color of the light bulb or flash the light so many times when an email come in from a specific person, etc. It is not just for email. It ties into many different services and apps. 

For your needs:

First search for IF by IFTTT in the app store and download it. 

Next create a recipe:

For the trigger set it to gmail for a new email based on your criteria you need.  

Finally add an action to call you based on that trigger and save the recipe.

This is based on you having the gmail app installed. If you are using the iPhone email app it might still be possible with the above solution but not 100% sure.