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Ticket 14554 was talking speciffically about an issue with System Restore Points. This is not your issue or, obviously, the same solution would also work for you. Would you mind please describing the exact issue you want me to help you with?

OK, then we're past the Akeeba Backup issue. What you are showing me in this screenshot looks like a problem with Admin Tools' Joomla! update. Which version of Admin Tools are you using?

OK, we can rule out the possibility of using an outdated release. Let's try the following, in this order:

• Try renaming the .htaccess file of your site to htaccess.bak while performing the Joomla! update. If that works, something in your .htaccess was blocking access to the administrator/components/com_admintools/restore.php script used to deliver the update.
• If that fails, try the integrated Joomla! updater (Components, Update Joomla!). This is a core component which is based on Admin Tools Core's code and uses a similar approach. If that fails, too, with an AJAX error then something on your server most likely doesn't allow direct access to its copy of restore.php.
• ...which leads us to test #3. Download the Joomla! update package directly from joomla.org and try to install it as an extension. If that works my guesses above are correct. If it doesn't work, you're in deep trouble. That would mean that the server doesn't allow you to run the post-upgrade scripts. In this case roll back from a backup, do the update on a local copy of your site and restore the (updated) copy of your site to your live server.

I wrote both components and they both work the same way. What can I say? Damn computers :)