If a nuclear power plant shuts down spontaneously during a solar storm, what about its cooling?

Answer

I am by no means an expert on nuclear power plants, but I dare say that the answer to your question actually lies in the question itself.

When the power surplus in the grid induced by a solar storm shuts down a nuclear power plant, it is because the security system of that nuclear power plant sets an alarm level. So it is not an external effect that was not taken into account at all, but rather the result of a protection against unexpected effects. It is hard for me to imagine that engineers who take the precaution to design a protection system against unexpected circumstances do not elaborate in terms of the processes to be carried out within the reactor itself.