Since the Admin Tools in Share are required to manage / modify groups (i.e. add someone to the admin group), there is no other, clean out-of-the-box way to make someone an admin.
In case you are using an authentication system other than alfrescoNtlm to authenticate users, you can use that subsystem's "defaultAdministratorUserNames" configuration property (in its *.properties file) to list the user names of all the users that should be considered admin if authenticated by that subsystem. By adding the user name you wish to make admin to that list, restarting your Alfresco server and then having that user login via that subsystem, you can effectively make them an admin without using the Admin Tools.... Though I reckon if you have access to reconfigure the server and restart it, you should also have access to the Admin Tools.
