Vibration, noise, and cavitation remain challenges for underwater propulsors. Besides blade design, a multi-stage structure is one of the ways to solve these problems. Based on the concept above, this paper proposes a co-rotating two-stage pump-jet propulsor. Due to the complex structure, the ability to capture inflow characteristics in numerical simulations would significantly affect its hydrodynamic performance results. Therefore, this paper investigates the influence of a numerical calculation method on prediction results of the two-stage pump-jet from the perspectives of mesh density, time step size, and turbulence model. The results show that differences in the rotors' inflow lead to varying convergence of their calculation results with changes in mesh density and time step size. It is also observed that the unsteady performance of both rotors is closely related to the periodic components in their inflow. A large time step size hinders capturing the forward rotor wake component in the rear rotor's inflow. The shear-stress transport k–ω model and improved delayed detached eddy simulation (IDDES) model are compared. The IDDES model can capture more detailed vortex structure in the flow field, resulting in broadband characteristics in its results and observable low-frequency differences between the two rotors. The research is helpful to the selection of the calculation method for multi-stage pump, tandem propeller, and other multi-stage rotary machinery.