As the scope on a rifle is mounted about an inch above the barrel, does this mean that the shooter needs to make this adjustment when zero-ing on a target?
No... if mounted correctly and sighted in for a given distance, say 100 yards, holding the sights (cross hairs or post) on the taget will result in hitting the target. Compensations do have to be made for wind or different distances than the one for which the rifle and scope were sighted.
dont forget though, when you have the scope mounted, it needs to be calibrated. then you can make slight adjustments on the range as needed. there are two adjustments on the scope, windage (left to right) and elevation (up and down). make these adjustments a click or two at a time.