I think “me time” is the way I usually see it.

I say, "me time". If I were to write it out, I might use the quotation marks depending on whom I'm talking to. The phrase is slang...a made-up word, but it basically means, "I need some time by myself" or "time alone" (quiet solitude...) It could also mean time to do something for yourself, like get a haircut, pedicure, go shopping, go for a walk, etc. Your first choice is correct, but nobody is going to correct you because it's not a real word. :-) (I'm writing casually here, and not focusing on perfect grammar, lol)

Like “Having some me time.” I haven’t seen it like ‘me-time’ before. I think it depends on the person and their creativity when it comes to writing. Does this help?