If I attack with a manifested (face down) card and my opponent declares blockers, is priority then passed to me so I can turn it face up for its mana cost, or must I turn it up after declaring attackers?
Yes, you get priority during the Declare Blockers step. Rule 509 describes how the Declare Blockers step progresses, and rule 509.5 says
Fifth, the active player gets priority. Players may cast spells and activate abilities.
You can turn a manifested card face up any time you have priority, so that includes during the Declare Blockers phase. Rule 701.31b says
Any time you have priority, you may turn a manifested permanent you control face up. This is a special action that doesn't use the stack (see rule 115.2b). To do this, show all players that the card representing that permanent is a creature card and what its mana cost is, pay that cost, then turn the permanent face up.