The C-band photocathode electron gun is designed. As one of important equipments for the linear injector of the Southern Advanced Photon Source. This paper discusses its RF and coupler design. The C-band electron gun has a working frequency of 5.712 GHz, a 3.6-cell structure, a π-mode acceleration mode, and it adopts a coaxial coupling method. The CST and Superfish codes are used to optimize the cavity microwave structure, reducing the surface electric field of the cavity to enhance the accelerating field strength and suppressing the multimode transmission. Furthermore, COMSOL Multiphysics is applied to analyze the cavity cooling system, reducing frequency drift caused by the cavity heating. Additionally, a water cooling design for the cavity ensures that the maximum temperature rise of the cavity is less than 20 ℃. Under an input power of 18.15 MW, the acceleration gradient of 180 MV/m on the cathode surface is achieved, the ratio of the acceleration gradient to the cathode surface electric field is approximately 0.93, and the cavity quality factor is greater than 10000. The design of the coupler suppresses the transmission of dipole and quadrupole modes, with the S₁₁ parameter being less than -40 dB.