Server side Drupal install as git local repo

Question

I have a Drupal install on a shared server and would like to do some version control on it, namely git.

I'm a beginner at Drupal and only just starting with git so I've become a bit overwhelmed with any instructions on the web for doing this, as they assume you're proficient with one or the other.

What I want my setup to be is

• The hosted Drupal install is my git's "local" repo.
• Make changes, commit
• Push to github

The problem is that as it is a shared server, I don't think I can install git. So, firstly, is this workflow possible without installing git on the server? And, if so, secondly, could somebody please point me in the right direction for setting this up.

As always, if somebody has a better way of doing it, please say!