If you get installation errors or a blank screen while attempting to install the component, please follow our component installation troubleshooting steps. Please note that even though the instructions in that document talk about Akeeba Backup, the same procedure is valid for any component installation in Joomla!.
The Admin Tools update is handled by Joomla! itself. If you have the Professional version you need to follow the instructions to enter the Download ID and reset the Joomla! update cache.
If you know there are updates to Admin Tools but you can still not see them please go to Extensions, Manage, Update and click the Update Sites in the sidebar. Find the Admin Tools update site and make sure it's published. Now go to Extensions, Manage, Update and click the Clear Cache button. Now click the Find Updates button again. You should see the updates.
Any other update issue has to do with Joomla! and/or your host. Please request help for updates in the Joomla! support forum or ask your host for assistance. Usually the problem is either a firewall or a configuration setting on your server which prevents Joomla! from downloading the update package. Do remember that you can always download the installation ZIP file of the latest version of our software and install it twice in a row, without uninstalling before or in between, to apply the update on your site. This and more ways to update our software are contained in the documentation of the software.