My play group doesn't play with Ambassador often, mostly due to the impression that it leads to degenerate games, so I don't have much experience with this card, and the question in the title is one that has always puzzled me. In the few times that I have tried it on an engine board, I've only picked it up when I had an extra buy and was reliably drawing most of my deck. This tends to not happen very early when the earlygame is defined by slow build-up and Ambassador tennis.
Of course, the problem with using Ambassador as a cursing card is that it requires 2 cards in order to effectively deal out the Curse and doesn't provide some other kind of benefit - i.e., you don't need as much support to fire a Witch at your opponent every turn, but you do need a good enough deck with other supporting components to pair Ambassador with Curse every turn. Which is why I always wait until I know this match-up is nearly guaranteed to buy the Curse; the potential momentum loss from missing a cursing turn is a little too scary for me.
That said, I've seen people pick up the Curse pretty early in some logs, maybe as early as after the 2nd reshuffle. Am I not playing this optimally? Certainly there's some degree of discretion required on the part of the player, but generally, when is the best time to pick up the Curse, if at all?