This is absolutely possible, but must be custom-built for your app :)
I'll assume that you're going to be using the Shopify recurring billing API.
You'll want to create a new page in the app that lists the available plans, features, prices, etc, and buttons to upgrade (and probably downgrade). When a user hits "upgrade" you'll simply create a new RecurringApplicationCharge and require them to accept it in order to have access to the premium features, and have some code to verify that only people with an active charge have access to those features.
If you have a "downgrade" action, you would simply cancel the current recurring application charge.
Hope this helps!