Updates are not so bad.
I downloaded 5.19 and 6.13 from drupal.org to /root.
Then I extracted both files creating the directories /root/drupal-5.19 and /root/drupal-6.13
Immediately I renamed /root/drupal-5.19/sites/default/settings.php to settings.php.org. If you forget this step you will wish you had a backup of settings.php.
Then I "su" to the appropriate user; cd to public_html/domain_name and
cp -a /root/drupal-x.yy/* .
This copies the updated Drupal core over the current version. I get away with this on minor upgrades, e.g. 5.18 to 5.19. Don't do it from 5.x to 6.x.
Next cd to sites/all/modules and
If you don't have drush, install it now. It will save you a lot of agony. If you have the old version, you have to remove all vestiges.
Test your upgrade and look at the status report.