This work presents an optimization study on the RF cavity geometry design and beam dynamics of an electron injector linac for the Korea Fourth-Generation Storage Ring (Korea-4GSR). It is challenging to achieve an optimal design for a linac system due to the nonlinear nature of accelerator system and the associated beam dynamics with numerous knobs to control. To address this challenge, we employed the Multi-Objective Genetic Algorithm (MOGA) for efficient optimization. Through the utilization of MOGA, we effectively optimized the electron linac for the Korea-4GSR. The optimized design yielded a horizontal normalized rms emittance of approximately 3.79 mm-mrad and an energy spread of about 307.37 keV, which met design requirements of the Korea-4GSR. This approach facilitated the design of an electron linac tailored for the Korea-4GSR with enhanced efficiency and effectiveness.