Cable-driven aerial manipulator is considered as a novel robotic system which contains an aircraft and a multiple degree-of-freedom manipulator. The cable-driven manipulator is a typical nonlinear system with extremely complex dynamics, which may lead to inadequate control performance when subjected to lumped disturbances. A disturbance rejection controller was proposed to enhance the tracking performance of the aerial manipulator in joint space faced water samples at drain mouth. This controller synthesized fast continuous nonsingular terminal sliding mode technique and linear extended state observer. Firstly, the structural design of the aerial manipulator was introduced and dynamics model considering flexible joint was established. Secondly, the linear extended state observer was used to estimate and compensate the lumped disturbances, and fast continuous nonsingular terminal sliding mode technique was applied to ensure the system states can converge in finite time and restrain the chattering of the control torques. Furthermore, stability of the controller was proved by Lyapunov theory. Lastly, the efficiency of the proposed controller was tested through visual simulation and ground water sampling experiments. The results showed that the proposed controller had faster converge, stronger robustness, higher precision, and greater performance of disturbance rejection compared with the other two controller, which can satisfy the needs of water samples.