Should there be a hyphen in the sentence, early '80s or should it be written without the hyphen between early and '80s?
If you're using 80s as a noun, it should not have a hyphen, because we don't normally join adjectives (early) to nouns (80s) with hyphens:
- He wrote that song in the early 80s.
If you're using early 80s as a compound adjective, a hyphen is optional, and (depending on context) can add clarity:
- That's an early-80s song.
I think when it's used as an adjective, you use a hyphen: "That song you just wrote sounds so early-80s." I've been taught to do that when I want a phrase to serve as an adjective. I see it done in a lot of writing.
Side note: About the placement or omission of the apostrophe - I'm not sure.
It should be like early 80's and I don't think any hyphen is needed.