If you stand up quickly, does the pressure in your carotid artery drop?

Yes but !! In normal young person not on any medication, standing up will have a very transient drop in systolic pressure but almost immediate reflex spasm in periphral arteries and increase diastolic pressure, so end result is increase pressure when standing. But sick patients with dehydration, low blood volume, taking medications that dilates arteries and others will lose this reflex and will drop bp.