I'm summarizing explanation from the following link:
Predestination Timeline Explained
The sequence of events in Jane/John/Barkeep/FizzleBomber is as follows:
Jane left at orphanage
Jane grows up, Jane meets John
Jane gives birth
Jane's baby is stolen
Jane becomes John
John meets Barkeep at the bar
John goes back in time to meet Jane, gets Jane pregnant
John leaves Jane
John learns that Barkeep is actually John from a future timeline
John goes ahead into the future
John becomes a temporal agent
John travels back in time to defuse a bomb, burns his face and returns
John becomes Barkeep (gets face graft)
Barkeep goes back to meet John at the Bar
Barkeep travels with John to get him to meet Jane
Barkeep makes one illegal jump to the defusal moment to stop the event, fails, watches John burn his face
Barkeep travels to the time when Jane's baby is born
Barkeep steals the baby and leaves it at the orphanage in the past
Barkeep travels to the time when John and Jane are together (on the bench), takes John to the future and leaves him there to become a temporal agent
Barkeep retires in 1975 but the time device doesn't decommission
Barkeep is sees a clue to the Fizzle Bomber's location by Mr. Robertson
Barkeep goes to this location, finds out that the Fizzle Bomber is an older version of himself
Barkeep kills Fizzle Bomber
Now, to answer your question, his last legal mission as a temporal agent is to steal the baby of Jane, take it back in time and leave it at the orphanage, followed by this he is to bring John to the future to let him become a temporal agent.
This is the last and vital mission because this event ensures his own existence and his joining the Temporal Bureau (which also ensures his own existence). This ensures there is no paradox.
It is the last mission because there is a limit on the number of allowed time leaps because prolonged time leaps causes the mind to have delusions.
Mr.Robertson mentions to Barkeep that the Fizzle Bomber is not his problem because he's working within the rules of the Bureau. Before Barkeep steals the baby, Mr. Robertson mentions that he doesn't like the bureaucracy that exists. So in secrecy he gives Barkeep clues to locate the Fizzle bomber. This is not a legal mission, this happens in secrecy post the legal retirement of Barkeep as a temporal agent.