You are mistaking prepared spells with adventure spells. The white Emeritus has a prepared spell, which requires the creature to be on the battlefield, and prepared, before being able to cast the prepared spell.

Do you know how creatures with prepared spells work? The creature has to be on the batlefield and get prepared. Only then can you cast the prepared spell. You cannot cast a prepared spell from hand as an alternative to casting the creature.

So to get the Emeritus of Woe's Swords to Plowshares spell, you have to get the Emeritus onto the battlefield, then you have to fulfil its prepared condition to get it prepared (meaning the ETB trigger has to resolve with an opponent having more creatures than you). And if it is prepared, you can cast the StP at any time, so long as the Emeritus stays prepared.