Speaking of screws by the way... your hooks could be connected to bolts that fit on a threaded rod. As you rotate the rod, it will travel the nut up and down the length of the shaft. The mechanism I showed you would still handle the reversing of this threaded rod. To eliminate what they call 'backlash" which in your case would mostly happen when switching directions, the "nut" usually consists of inner ball bearings which roll inside the groves. Your pitch and rotation speed will determine how fast the movement moves up and down your roll. This application would not require a fine pitch.