The thing is, that's actually the easy bit - use apt-get upgrade on say Ubuntu lts and you will very rarely see problems with your upgrade of software like Postgres, apache, ssh or nginx - it's so widely used that you're unlikely to run into issues, and upgrading takes a few minutes every few months.

The thing you need to test then deploy are upgrades to your language and framework/app, and those present exactly the same problem on heroku or on say a vps. You can't just redeploy your app without testing on a new config, and deploys to heroku are no easier or harder than deploys to your own server once set up.