Used by the Vizier, who appeared adept in the ability to control the Sands of Time with spells, the Staff of Time can amplify the powers of the Sands of Time, using it to create small storms to disable or disorient his opponents.
In addition, he appears able to create duplicates of himself. The Vizier cannot simply use the magic and powers that the Staff can command unless he recites some kind of incantation. Like the Medallion of Time, the Staff protected him from the effects of the Sands of Time.
It is assumed that the Staff of Time was created in conjunction with the Dagger and Medallion of Time, presumably by Kaileena herself (though the latter is never outright confirmed in The Sands of Time Trilogy).
Relocation to India
When the Maharajah visited the Island of Time, he obtained all three known artifacts along with the Hourglass of Time. The Vizier, the Maharajah's adviser, likely maintained or gained possession of the Staff while the Medallion was given to his daughter, Farah and the Hourglass and Dagger were hidden away in the treasure vault.
Sands of Time
The Staff of Time protected the Vizier from by the Sands of Time, when the Prince released them. The Vizier seemed able to control the Sands of Time and the Sand Creatures it created. The Sand Vultures brought the Hourglass of Time to top of the Tower of Dawn, and in a vision, the Prince witnesses a Red and Blue Sand Guard standing on either side him.
When the Prince triggered the Grand Rewind and infiltrated the Maharajah's palace to warn Farah of, the Vizier uses the staff as a weapon, attempting to kill the Prince with it. The Vizier used the Staff to create identical sand copies of himself to combat the Prince for him, controlling them like puppets, while forming a barrier of magic around himself. But with the death of each "clone" the Vizier became exhausted to the point where he could no longer fight and was forced to fight in vain against the Prince.
The Two Thrones
The Vizier holds the staff when he stabs Kaileena. He drops it after he stabs himself with the Dagger of Time. The Staff was likely destroyed in the chaos of the crumbling surface of atop the palace in Babylon after the Vizier gained immortality as Zurvan.