Ponerse is a really common way to refer to a person's change in mood or color or expression of the face. You can use it in present tense, like you have done, but it is more common in the preterit.
As people have said, ponerse como un tomate can refer to turning red in embarrassment, or from a sunburn or from anger.
To steal from Antonio:
Cuando se dio cuenta de que no debió hacer ese comentario, se puso rojo como un tomate.
After he realized he shouldn't have said that, he got/turned/became red as a tomato.
One last thing, I don't think your sentence is that natural. Here are some options:
“Me pongo como un tomate cuando me siento en la sauna por mucho tiempo.”
"Me pongo como un tomate tras un rato en la sauna."
I'm almost certain you need 'la sauna' here. 'Una' sounds really weird. You are talking about all saunas in general. However, if you made a qualifying statement about the sauna, then you would use "una." If this is wrong, I hope someone corrects me.
Me siento en la sauna para sudar.
I sit in saunas to sweat.
Me siento en una sauna a 80 °C para sudar.
I sit in a sauna at 80 °C to sweat.