Reaching SQL Server AlwaysOn HA listener from the Internet
It's generally considered a really bad idea to make your database accessible from the internet. 

Making the database accessible from the public internet is a HUGE security risk. It's easy to port-scan and find publicly accessible databases. Even if you run on a non-standard port, it's relatively trivial to discover the SQL Server instance that is internet-facing. Your database would be available to anyone to exploit. It's likely that you'd be victim of every zero-day exploit, in addition to brute force attacks. Eventually, your data will be stolen or destroyed. 

Take a look at shodan.io to see how easy it is to discover databases on the public internet. You will essentially have a big sign saying "HACK ME" hanging on your IP address. 

Ideally, you would have a web service that the application would interact with. That public web service would be the only thing that connects to the (non-public, firewalled) database. Doing this would require re-architecting your application to use the web service, rather than a direct client-server database connection. If you can't re-architect the application, then VPN is the only secure solution.