Hi Romke,
it could be helpful if you take all files you changed, but also their original from JEM 2.1.3 and those from JEM 2.2.1. Then you can make a three-way-diff to create your modifications corresponding to 2.2.1.
Another way could be to make a "dry run" by installing a webserver (e.g. xampp) on a local pc where you can restore your full backup and try to update JEM without risk. (I should like to point out that local webservers like xampp may be configured fully unsecure by default to make developer's life easier so you have to ensure nobody can access your local pc to protect the secrets of your site's users)
Additionally you could export all events, categories, and venues via JEM backend. So on trouble you could import these data. Generally JEM is designed to be able to update from any official JEM release. But it could be possible that the server is too slow and/or the timeout is too fast to handle all the adaptions in time.
I'm very optimistic updating JEM is smart and without trouble - but I can't give a guarantee.