A series of superconducting insert coils (ICs) made of different materials has been tested since 2000 at JAEA Naka in the bore of the central solenoid model coil at fields up to 13 T and currents up to several tens of kA, fully representative of the ITER operating conditions. Here we focus on the 2015 test of the presently last IC of the series, the central solenoid (CS) insert coil, which was aimed at confirming the performance and properties of the Nb3Sn conductor, manufactured in Japan and used to wind the ITER CS modules in the US. As typical for these large scale applications, the cooldown (CD) from ambient to supercritical He temperature may take a long time, of the order of several weeks, so that it should be useful, also in the perspective of future IC tests, to optimize it. To that purpose, a comprehensive CD model implemented in the 4C code is developed and presented in this paper. The model is validated against the experimental data of an actual CD scenario, showing a very good agreement between simulation and measurements, from 300 to 4.5 K. The maximum temperature difference across the coil, which can only be roughly estimated from the measurements, is then extracted from the results of the simulation and shown to be much larger than the maximum value of 50 K, prescribed on the basis of the allowable thermal stress on the materials. An optimized CD scenario is finally designed using the model for the initial phase of the CD between 300 and 80 K, which allows reducing the needed time by similar to 20%, while still satisfying the major constraints. Recommendations are also given for a better location/choice of the thermometers to be used for the monitoring of the maximum temperature difference across the coil.