How can I get the content of a mini panel that is limited to admin-only access using a scheduled rules component?
The problem seems to be an anonymous user triggering the component and since you want an admin user to run the component, you will have to impersonate the admin user.

Try this code inside your components to impersonate the admin user :

global $user;
// Save the current user
$current_user = $user;
// Save user session
$current_session = drupal_save_session();
// From now on we don't want to save the session, since it would be unsafe
drupal_save_session(FALSE);
// We impersonate admin user
$user = user_load(1);
// Do wathever needs to be done.
// ...
// Now let's restore the triggering user and session.
$user = $current_user;
drupal_save_session($current_session);