I don't think even being able to survive getting near the sun makes sense. In real life you would get vaporized long before you reached any matter just by the extreme levels of radiation. And I don't see shields helping much at all, after all our shields can be eliminated by simple weapon's fire. Or lasers which are just concentrated bursts of em radiation, except magnitudes greater.
Really being able to enter a sun just stretches any suspension of disbelief out the window and strangles puppies in the street.
How fast will the ship heat up?
How is the heat IRL distributed to a ship near the sun
Is it possible to use materials which isolate in one direction but are thermal-conductive in the other to cool down faster?
Is it possible to regain energy from heat? If we can produce heat with power, shouldn't some future device not make it work (efficiently) also in the other direction too?
I think ships should be especially built for this. These may require some investment in mass (1-3%) and 85% of the mass in parts with currently no active function (hulls, metal grills, ...)
Other things you may want to ask, how much energy to escape the gravity of the sun? After all it is a very large mass (at least from our POV).
If you were to somehow enter the sun and encountered non-ionized gas would you be able to withstand the pressure.
Materials that are thermally conductive one way would probably be chemically based I guess, a reversible reaction in which the endothermic possess is favoured greatly. Perhaps for starmade we could just invent some almost-but-not-really-unobtainium plating for ships.
I don't think making energy from heat DIRECTLY is possible, everything kind of tends towards disorder so you have difficulty making something useful like electrical energy from heat without making a bunch of useless kinetic(pressure against container walls). Although considering how hot the sun is you probably could make decent energy. You might have issues venting the heat in space since you can only lose heat by radiation... and that doesn't work when surrounded by IR sources hotter than you.
But I digress I'm kind of being lead off by my dislike of the idea.
I think neon's last point could probably be an interesting mechanic. Involving power tanks and heat-> power recievers. Namely you turn off your current generators and switch to the heat ones, now as long as you stop your power from topping out (so the generators can continue to decrease ambient heat in the ship) you will suffer no/minimal damage.
"How heat is distributed to s ship near the sun" Only thing that can transfer heat near the sun is radiation, once you get to touching actual matter from the sun you get superheated gas and ionized gas. Pretty hot at 1 million kelvin in the corona(at a minimum) then cooling down to 20000k in the upper chromosphere. It only gets cooler from there although not sure we would get that far.
Could think of other things but I'm getting tired. Night all.