does wordpress auto update work without a cron job?

The latest versions of WordPress will update automatically. AKA. you do absolutely nothing and then at random times you’ll receive an email telling you it has self updated. This does not require any cronjobs etc. As for setting this up, it may or may not require FTP credentials (can’t recall off the top of my head).

edit: auto update will only work if FTP credentials don’t need to be entered manually.