Updating Drupal modules via git on remote server
In my opinion the workflow you mentioned look good. You just need to make sure you have a latest database backup in case you need to revert back. 
Have you looked into https://getpantheon.com/ it is free for drupal development. Request an invitation code at https://getpantheon.com/request-code Best thing about pantheon is it provides dev test and live sites, daily, weekly, monthly and on demand backups. Push code from dev to test and live, Pull database and files from live to test and dev with just push of a button.