A container crane mounted on a pontoon is utilized to transfer containers to smaller ships when a large container ship cannot reach the shallow water port. The shipboard container is considered as an underactuated system having complicated kinematic constraints and hysteretic nonlinearities, with only two actuators to conduct simultaneous tasks: tracking the trolley to destination, lifting the container to the desired cable length, and suppressing the axial container oscillations and container swing. Parameter variations, wave-induced motions of the ship, wind disturbance, and nonlinearities remain challenges for control of floating container cranes. To deal with these problems, this study presents the design of two nonlinear robust controllers, taking into account the influence of the output hysteresis, and using velocity feedback from a state observer. Control performance of the proposed controllers is verified in both simulation and experiments. Together with consistently stabilizing outputs, the proposed control approach well rejects hysteresis and disturbance.