As the demand for eco-friendly energy increases, hydrogen energy and liquid hydrogen storage technologies are being developed as an alternative. Hydrogen has a lower liquefaction point and higher thermal conductivity than nitrogen or neon used in general cryogenic systems. Therefore, the application of hydrogen to cryogenic systems can increase efficiency and stability. This paper describes the design and analysis of a cryogenic cooling system for an electric propulsion system using liquid hydrogen as a refrigerant and energy source. The proposed aviation propulsion system (APS) consists of a hydrogen fuel cell, a battery, a power distribution system, and a motor. For a lab-scale 5 kW superconducting motor using a 2G high-temperature superconducting (HTS) wire, the HTS motor and cooling system were analyzed for electromagnetic and thermal characteristics using a finite element method-based analysis program. The liquid hydrogen-based cooling system consists of a pre-cooling system, a hydrogen liquefaction system, and an HTS coil cooling system. Based on the thermal load analysis results of the HTS coil, the target temperature for hydrogen gas pre-cooling, the number of buffer layers, and the cryo-cooler capacity were selected to minimize the thermal load of the hydrogen liquefaction system. As a result, the hydrogen was stably liquefied, and the temperature of the HTS coil corresponding to the thermal load of the designed lab-scale HTS motor was maintained at 30 K.