My approach to this problem is to employ precisely the solution you have suggested about automatically turning the VM on.
Firstly, I tend to use the daily Auto Shutdown as a fall back in case i forget to turn the VM off when I am using it. (Say at 7pm).
Secondly, I configure scheduled updates on the VM which occur at 5pm on a Sunday (actually i tended to choose Friday, but pick whatever day works best for you. I just prefer to pick a day when the VM would usually be on, to avoid starting it purely for patching).
Lastly, I leverage automation to start the VM up - just in case I hadn't been using the VM that day.