Word for sarcasm/facetiousness where the core statement is true

There's a word custom-tailored to this very situation:

Irony

a form of humor in which you use words to express the opposite of what the words really mean.

_{Macmillan, sense 1}

If you're asking for types of humor, I'd go with cynicism, irony, or mockery.

cynicism

Person A: I think it's going to be hot and dry today.

Person B: It's not like we're in the desert or anything.

irony

Person A: Hey, we can come back from a 35-0 deficit.

Person B: Yeah, sure, like you're going to tell me about the one time it happened before. I mean, what the heck, it is in Denver, after all.

mockery

Person A: That's a nice zit you have.

Person B: Thanks, I just had it stuffed.

tongue in cheek

When someone speaks tongue-in-cheek, that means they're joking and kidding.

He always speaks tongue-in-cheek, he never takes things seriously.

by Fangsta March 18, 2003

Urban Dictionary