We propose a simple hopping mechanism using vibration of a two-degrees-of-freedom (2-DOF) system for a fast stair-climbing robot. The robot, consisting of two bodies connected by springs and a wire, hops by releasing energy stored in springs and travels quickly using wheels mounted on its lower body. The trajectories of bodies during hopping change based on mechanical design parameters such as reduced mass of the two bodies, the mass ratio between the upper and lower bodies, and spring constant, and control parameters such as initial contraction of the spring and wire tension. This property allows the robot to quickly and economically climb stairs and land softly without complex control. In this paper, we propose a mathematical model of the robot and investigate required tread length for continuous hopping to climb a flight of stairs. Furthermore, we demonstrate fast stair-climbing and soft landing for a flight of stairs in experiments.