To solve the position tracking control problem for a class of manipulator systems with modeling errors, external disturbances, and time-varying asymmetric output constraints, a fixed-time backstepping adaptive optimal control method is proposed. Firstly, a fixed-time disturbance observer is designed to quickly and accurately estimate the modeling errors and external disturbances of the robotic arm system. Secondly, the fixed-time controller design considering the output constraints is accomplished by combining the backstepping method, a new obstacle Lyapunov function, and a rolling adaptive law, while the command filter and the switching function are used to solve the “differential explosion” and “the singularity of the virtual control quantity” in the controller derivation process. Then, the improved northern goshawk (ING) algorithm is used to optimally tune the control parameters of the robotic arm system to further improve the convergence speed and steady-state accuracy of the system. The theoretical analysis shows that the tracking error of the system can converge to a small neighborhood close to zero in a fixed time, and always satisfy the output constraints. Finally, the effectiveness of the proposed method is verified by a comparative simulation study of the PUMA 560 robotic arm.