For talking about attempting something:
He succeeded __ the third attempt.
According to Google, the preposition of choice is "on". Would "in" cause outrage?
While prepositional use is often idiomatic, in this case there is a logic to on instead of in.
On is often used to refer to a discrete, singularly identified period of time, even when the time described has a significant duration.
When you are seeking to connote that the time identification is not meant to be a singularity but rather over an extended period or at an unspecified time, you might use in.
Your sentence seems to seek to convey an outcome tied to a discrete, singular activity in a series of discrete activities.
What is important is the distinction between the singular event, the third, and the two previous singular events.
However, there might be might be circumstances where the place within the duration of the activity is of more interreset, and in would be appropriate.