The self-driving of water droplets on textured substrate is momentous for understanding the directional transport of water on biological surface. In this paper, a novel design of monolayer graphene-covered wedge-shaped copper substrate (GWCS) is put forward to realize the directional and ultrafast spontaneous driving of water droplets from the tip of the wedge-shaped substrate to the wide end. The self-driving behaviors of water droplets on GWCS are studied by classical molecular dynamics. The results show that the maximum spontaneous driving velocity of water droplet driven by surface wetting gradient and Young-Laplace pressure gradient can reach 73.8 m/s. The law of energy variation during the whole droplet self-driving on GWCS indicates that there is a competitive relationship between the potential energy of water droplet and the interaction energy between the droplet and GWCS, i.e. the interaction energy between water droplet and GWCS is partially converted into the potential energy of water droplet. The relationship of the maximum displacement of water droplet with the wedge angle, wettability of GWCS and the droplet surface tension is proposed in theory, and the influence of the discontinuous linear increase of the wedge-shaped substrate width on a nanoscale on the self-driving is analyzed and used to explain the little difference between the theoretical and simulation results. Furthermore, a smaller droplet is easier to obtain larger spontaneous driving velocity, and the influence of long-distance decelerating motion of high-speed small droplet on the non-wetting gradient substrate on the droplet displacement law is clarified. Finally, the mechanism of graphene suspended on both sides of the wedge-shaped copper structure to enhance the droplet transport efficiency is determined. The results will have theoretical significance in designing the functional texture surface covered by monolayer graphene to realize droplet self-driving.