If the epic monster needs a certain item to be hurt, would advancing the mystery(MoM card) when you do not have the said item still work?

Advancing is not solving, so, I'd say no (still, don't have the final edition of the game, so, I don't know the final wording).

Wrong answer, brilliantly corrected by Randymd in the post below

Yes, it would work.

Page 6 in the Eldrith Horror rule book: "If the active mystery requires an epic monster to be defeated, place two health on the card. The epic monster's toughness is reduced by one for each health on the active mystery."

So technically, you're not dealing damage to the monster. You're reducing the monster's toughness instead, which basically amounts to the same thing.

Brilliant catch, kudos