I assume that you have built a steady state model. Ideally you need to build a dynamic model, as the situation changes with time. Initially, the tanker is empty with a pressure of 0.2 psi. As the natural gas flows into the tanker, the big pressure difference will result in a large reduction in temperature (Joules-Thompson Effect). As the tanker fills, this pressure differential (and drop in temperature) is reduced. The speed at which this happens will depend on the natural gas flowrate and the volume of the tanker.
If you can't build a dynamic model, I suggest that you try the following approach:
1. Model the system in steady state with a tanker pressure of 0.2 psi - calculate the inlet temperature
2. Model the system in steady state again, but at a tanker pressure of 300 psi - calculate the inlet temperature.
3. Assuming that there is no heat transfer to and from the outside (adiabatic conditions), as a first approximation the temperature in the tanker will be half way between the two calculated temperatures.
In addition, there are a couple of points worth noting. Firstly, at pressure differential of nearly 600 psi, the flow into the vessel will be choked (you can ask you professor about this). Secondly, at these very low temperatures, the materials of construction become important .
If you need any further help, please feel free to contact me.