Write down everything you come across when updating locally and then use that as the basis for a script/procedure to update production? I don't mean to come across as flippant - really :-) But that's about the best you can do without using something like laravel shift - which is really just outsourcing that process I guess.
If you're on linux/os-x there's a handy cli tool called 'script' which can be helpful, for instance :
script update-this-laravel.log
That will start another shell and everything you type or that is displayed back will be recorded - just exit the shell once you're done and you can go back through later and see everything you did. It's handy when you're not quite sure what might be broken or you need to do.