(No drush!?!? #inconceivable)
With all of the fancy ways to upgrade Drupal core floating around: drush, git, diff, patching. What if you only have FTP?
As always, the first steps are to a) plan, b) backup, and c) be flexible.
It always helps to have a plan. Write it down, think it through, practice on something other than your production site.
The basics for updating a Drupal minor version (eg., 6.20 to 6.22) are that you need to save your /sites folder and anything that you've made changes to outside of the /sites folder. Usually those changes might be to .htaccess or robots.txt.