The effect of Captivating Vampire’s activated ability has no duration. You retain control of the affected creature until the game ends, the creature leaves the battlefield, or a later effect causes someone else to gain control of it. It doesn’t matter whether Captivating Vampire remains on the battlefield. Similarly, the affected creature remains a Vampire in addition to its other types until the game ends, the creature leaves the battlefield, or a later effect changes its types or subtypes.

Since Captivating Vampire’s activated ability doesn’t have a tap symbol in its cost, you can tap a Vampire (including Captivating Vampire itself) that hasn’t been under your control since your most recent turn began to pay the cost.

Gaining control of a creature doesn’t cause you gain control of any Auras or Equipment attached to it.