Once a Captain has been promoted to Admiral, what becomes their official duty? Do Admirals have certain predetermined Captains and ships under their command, or do they get orders from higher up? What's an Admiral's day-to-day job?
Admiral commands a fleet of ships (as opposed to a single ship) with help of other admirals whether senior or junior (Fleet Admiral, Rear Admiral etc). This is similar to real world military ranks. He can monitor captains, suspend captains, assign new captain to a ship & send ship(s) to a particular mission.
An Admiral's day-to-day jobs are mostly official paperworks, but he can actively command a ship if he wish or its important for a mission (The command of vessel is automatically transferred to superior officer on-board as we've seen in the canon).
Memory Alpha has a dedicated article on this. Maybe, you should look at it.
From the ufstarfleet.org wiki the description is quite extensive and seems to parallel much of what a current US Navy (or any other Navy) Admiral performs as his duty. The description from the wiki;
The role of an admiral is highly diverse, and the duties are wide ranging and often complex. These vary from the mundane, such as attending diplomatic functions, lecturing at Starfleet Academy, and inspecting starships prior to launch, to being forced to put the lives of Starfleet officers on the line when authorizing vessels or personnel to enter hazardous situations.
It is a position that combines duty to the service with a requirement for skills of a political nature. Admirals are also granted the authority to decommission starships that are deemed past their prime or irreparable, as demonstrated by Admiral Morrow’s orders following the return of the USS Enterprise NCC-1701 to Spacedock in 2285.
In addition, there are many admirals who supplement their duties by maintaining a posting as commander of a starbase or other non-starship position, such as Admiral Marcus Holt, who commands Deep Space Three. Of similar note is Admiral Brand, who holds the position of Superintendent of Starfleet Academy on Earth.
The rank of admiral, particularly in times of peace, is primarily a desk job. The officer in question is often stationed at a Starfleet facility on Earth or elsewhere in the United Federation of Planets. For some, this can be considered a detrimental aspect of the position, as James T Kirk discovered following his promotion. Kirk’s duties were many and varied, but he felt stagnant ‘flying a desk’ at Starfleet Headquarters in San Francisco. He jumped at the chance to take command of the refitted Enterprise when the V’Ger probe approached Earth in 2271.
There are of course other circumstances that occasionally demand the presence of a member of the admiralty. For example, an admiral may be placed in an extremely hostile situation, where a high ranking officer is needed to coordinate starship deployment and battlefield strategy, as shown by the actions of Admiral JP Hanson and Admiral Hayes during the Borg offensives of 2366 and 2373.
An admiral may also be assigned to oversee a delicate operation in a distant region of space, such as Admiral Matthew Dougherty’s supervision of the joint Starfleet and Sona mission in the briar Patch. During Starfleet missions of a sensitive nature, or operations conducted under wartime conditions, an admiral is often posted to a facility ‘in the field’ from which they command and control several vessels; during the Dominion War, Admiral Bill Ross handled fleet dispersal and mission tasking from Starbase 375. Ross’s position offers a typical example of such a posting; his command includes vessels such as the USS Defiant NX-74205 and ships from the Seventh Tactical Wing.
And it continues on, but I leave that for you to read for yourself. Link: Role%26_Duties">http://www.ufstarfleet.org/wiki/index.php?title=Admiral:Role%26_Duties