Are you sure that the 'Admin site' role is actually an administrator
role? It seems like one of your users is a real administrator while the
other is not, since access restrictions are 'bypassed' for real
administrators. You might want to check whether there is another real
administrator role in /admin/people/permissions/roles (the default
administrator role is 'administrator).
To access the pages, setting view permissions for the Admin site rol in
the access control module should be sufficient.
In order to let the Admin site role see/edit admin settings, you should
set the right permissions in People => Permissions => Administer xxx for
the Admin site role.