An end-effector position tracking control task for an aerial manipulator is usually constituted by two subtasks. The first is motion control, and the second is coordinate control so that the end-effector of the aerial manipulator can precisely track the given trajectory. This paper proposes a novel end-effector position tracking control approach for the aerial manipulator with a lightweight manipulator to achieve these two subtasks. The motion control of the aerial manipulator is solved by a partially coupled approach and divided into a multirotor controller and a manipulator controller. The multirotor controller is designed by the adaptive neural network control, while joints of the manipulator are steered by PID controllers. By resorting to radial basis function neural networks with adaptive weight estimation laws, the dynamic coupling between the multirotor and the manipulator can be compensated in real time. With the support of Lyapunov stability criteria, it is proved that the desired trajectories can be boundedly tracked by the multirotor under the proposed controller. Then, a new coordinate control method is proposed based on the linear model predictive control method. This method ensures that the solution satisfies physical limits of the aerial manipulator and can be executed in real time. Simulations demonstrate that the proposed motion controller significantly outperforms a baseline nonlinear motion controller in the simulation cases. Besides, comparisons among the proposed coordinate control method and traditional methods are simulated to demonstrate effectiveness and performance.