This paper studies the automatic carrier landing controller design problem with output constraint and fixed-time convergence requirements. The proposed flight controller is separated into guidance, attitude control, and approach power compensation system. In each submodule, universal barrier Lyapunov function and fixed-time control scheme are applied in the controller design. In addition, adaptive estimation method is incorporated to improve the robustness to the external disturbances. The proposed controller ensures the position tracking error, attitude tracking error, and angle of attack tracking error not violate the predesigned boundaries, while the tracking errors will converge to a small set around zero in fixed-time. The output constraint and fixed-time convergence properties of the proposed controller improve the landing security to a narrow flight deck in a limited operation time. The simulation results verify the proposed controller in automatic carrier landing mission.