My son is tall and skinny he is medium when it comes to shirts but I don’t want his shirt to be to big I want it just right on him he is 15yrs old

I'm 5'9 and wear a medium as well , I ordered a medium and it fits just right

If My sons a Men’s size medium, should I get a size up?

Yes. The shirt is fitted but stretchable. But to feel comfortable get one size up. Great material though

Is the white shirt solid or cheap somewhat see through material? (Can you wear it without a t shirt)

I got a black shirt, so "see through" wasn't an issue. It was shiny, cheap-looking material like something to wear to a club. If you get a white shirt, wear a t-shirt.

I tend to get overheated. Is this shirt made from material that allows the torso to remain cool?

The shirt is very comfortable and it was worn when the temperature was 100 and in full suit and no one complained of being hot and it hasn't had to be laundered.