Joomla!'s extension update system is known to have sporadic issues which can result in a partial update being effected on your site. In this case you will indeed get locked out of your site and / or get a white page or Internal Server Error error. This kind of update issue is most likely to occur when there are significant differences between the installed and the new version of the extension which is indeed the case with updating Admin Tools 3.x to 4.x.
Don't worry, though, it's something you can easily work around if you follow our instructions.
First try deleting the plugins/system/admintools directory from your site using FTP/SFTP or your hosting control panel's file manager feature. This should allow you to regain access to your site's back-end.
If you can't access your site after renaming the folder please contact your host and ask them to reset the code cache they have enabled on your site.
Now that you can access your site's back-end go to Extensions, Manage and install Admin Tools twice in a row, without uninstalling it in before or in between. This will force Joomla! to install all the files it was previously missing. Since Admin Tools is now properly updated you can access your site just fine and protect it with Admin Tools.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!