So he's saying is that the mail is a priority and has to be done by a certain time? That seems reasonable to me from an over-all point of view because the delay of mail distribution causes a cascade of delays.
I suggest that you create an operational policy on how this will be done with times and make sure everyone agress it, including the people the admin reports to. Set out statements so people can see how they might affect their own day-to-day work.
1. The sorting of mail must be done immediately/by 9.30am every morning by the responsible person.
2. The responsibility for mail sorting will be assigned to admins on an equal rotational basis.
3. In event of any person not being able to fulfill their mail duty, the task moves to the next person on the rota until the first person is able to return to the duty.
It's important that the people you report to understand that this is what will happen so that if your boss has an emergency (or even wants his coffee and danish fetching

) he will know that he will have to wait twenty minutes for your attention. Otherwise all bosses will start pulling rank and demanding that their admin is exempt because they themselves are too important (which is rarely true).
HTH
Bethalize
Peer Moderator