The aerial manipulator, designed for complex aerial tasks, encounters multifaceted operational environments influenced by various internal and external disturbances. This paper introduces an adaptive neural network backstepping control technique fortified with coupling disturbance compensation to enhance the resilience of the aerial manipulator against these disturbances. Firstly, we propose a cutting-edge coupling disturbance feedforward compensator based on variable inertia parameters, which offers precise and prompt compensation for significant internal coupling disturbances without needing external sensors or alternative disturbance estimation techniques. Subsequently, radial basis function neural networks with an online adaptive weight updating mechanism are designed to estimate and counteract lumped disturbances stemming from unmodeled dynamics, uncertainties, and external factors in real-time. Utilizing the Lyapunov stability criteria, we validate that the aerial manipulator can reliably track desired trajectories under our proposed controller. Experimental results and simulations further underscore the effectiveness and superiority of our control approach.