This paper proposes a controller design method with appointed-time performance based on backstepping method to address the cruise missile attitude control problem. Firstly, a new predefined-time control law is presented, followed by the design of a prescribed performance function, a novel type of error transformation equation is established, and a controller for the transformed system employing backstepping based on this predefined-time control theory. Then the stability of the predefined-time of the designed controller is proved based on Lyapunov stability theory. By further derivation, it is demonstrated to have appointed-time performance. Finally, numerical simulation of the missile's longitudinal profile is performed in conjunction with the constructed controller to ensure that the controller is performing efficiently. The results show that the control law established in this study may allow the tracking error to reach appointed-time convergence while satisfying the predefined-time performance, enabling rapid and precise tracking of the guidance instruction.