To address the problem of high-precision attitude tracking control for spacecraft, this paper designs a class of prescribed-time control laws based on periodic delayed feedback theory. The designed control laws can drive the actual spacecraft attitude to track the target attitude within any prescribed time while avoiding the singularity issues of the controller. Furthermore, building upon the designed periodic delayed sliding mode control laws, a Prescribed-time Extended State Observer (PTESO) is introduced to achieve online observation of unknown disturbances. A composite control law integrating both components is then proposed. This composite prescribed-time control law can effectively eliminate the chattering phenomenon inherent in sliding mode control. Simulation results demonstrate that the proposed control laws can successfully complete the spacecraft attitude tracking control task within the prescribed time and exhibits strong robustness against external disturbances.