Heroku is nice, most rails tutorials recommend it.
That being said, I wonder what's stopping you from setting up a machine on AWS or Digital Ocean. Once you have the initial setup ready, it's a matter of triggering a .sh file using webhooks directly from your version control.
You can also have your version control hooked up to say Codeship, Shippable or Buddy (all of them have excellent tutorials to get started, DevOps won't be challenging at all if you let these guys handle it for you) and they will take care of triggering your ./sh after running your tests. Let me know if you need assistance with any of the things I said - happy to help! :-)