Servo press technology is gaining attention because its punch motion curve offers greater formability than that of a conventional stamping press. This study investigated the effect of punch motion curves on the circular cup extrusion–drawing process. Various punch motion curves were analyzed, and the optimal curve for application was determined. Both the extrusion–drawing process and spring back of U-shaped sheet metal were investigated. In the circular cup extrusion–drawing process, the punch motion curve of a conventional stamping press (Case A) and three punch motion curves of a servo press (Cases B–D, the strokes of which differed from that of Case A by 0.5, 1.5, and 2.5 mm, respectively) were compared, particularly regarding the effect of the coefficient of friction on the circular cup extrusion–drawing process. The simulation analysis was performed using the software program DEFORM. A set of simulated parameters were compared with experimental results. The formability, cup shape, cup height, cup thickness, extrusion force–displacement curve, stress distribution, and strain distribution were analyzed for the design of the die required. Additionally, experimental and simulation results were compared to determine the reliability and precision of the DEFORM simulations. The results indicated that the conventional punch motion curve resulted in a shorter cup, greater stress, greater strain, and the need for a greater extrusion force. By contrast, the servo punch motion curves resulted in taller cups, less stress, and less strain. The findings can serve as a reference for the development of servo presses.