This is a key topic, and one that isn’t necessarily going to yield to my attempts at an answer. An archmage is and can be many things. These are not (yet) presented in any particular order.
S/he is the organizational principle of the fellowship. Working with the fellowship, s/he determines goals and strategies, practices that are suited to the fellowship’s personality and play style, ground rules, and more. This is a key role, but is often a bit more subtle, as the more obvious communication and support staff functions tend to be the most visible (and easier) tasks.
S/he is the principal communicator. While you should have mages and other fellowship members helping you out, it is your job as archmage to stay in communication with the fellowship. Make sure they know what’s going on; expectations you might have; goals, and progress toward those goals; that you notice when they meet a major milestone; and any one of a number of things. While you communicate these things, large and small, you also communicate your competence and that you care for the fellowship – two key things that the fellows need to know for your fellowship to succeed.
S/he is the cheerleader. Cheer people on, notice when they meet a milestone, walk them through the tough parts, call out how well the fellowship has done in a tourney – all those little successes need to be known, especially if the fellowship is full of small cities and/or new players. Challenge them…. then cheer them on.
S/he is the support staff. It’s your job to see that housekeeping gets done. Is someone making a habit of not visiting? Talk to him. Someone leave the fellowship? Put on your recruiter hat. Is your tournament mage sick? Best see to it that you get the preparation message out for the upcoming tournament. You get the idea – these are the “duties as assigned”.
As an archmage, it’s important to remember that you do not have a captive audience – any player can opt out with a few clicks. If you become dictatorial, people will leave. If you’re not a master of the art of persuasion and people management, this is a good opportunity to work on your skills.