The flying around monitoring task of space tumbling target is one of the key links of its on-orbit service. Considering the practical constraints of system inertia uncertainty, external disturbance, actuator saturation, and fault in engineering practice, a robust composite controller based on radial basis function (RBF) neural network is proposed. First, in a new line of sight rotation (RLOS) coordinate system, the relative attitude kinematics and dynamics equations between the tracker and tumbling target based on the error quaternion are established; second, the RBF neural network is used to estimate the additive and multiplicative faults of the system, and the fast nonsingular terminal sliding mode surface (FNTSMS) is combined with the active disturbance rejection control (ADRC) technology to design a finite-time faulttolerant control (FTC) strategy with high accuracy, strong robustness, and anti-saturation based on the RBF neural network. It is proven that the designed robust fault-tolerant controller can ensure that the system state error converges to a small region containing the origin in a limited time under the Lyapunov framework. Finally, the effectiveness and superiority of the control strategy were verified by numerical simulation.