Today's batting order hasn't been announced yet.
The manager rearranged the batting order.
The manager set today's batting order.
His turn at bat came around in the ninth inning.
The manager changed the batting order partway through the game.
He was entrusted with the cleanup (4th) spot in the batting order.
打順 is the order batters hit in, while スタメン (starting members) just means who starts the game, without specifying batting order.
打席 refers to a single turn at bat or the batter's box, while 打順 refers to the fixed order in which players bat.
打線 refers to the batting lineup as a whole unit (its overall strength or flow), while 打順 refers to an individual player's specific position within that order.
Compound of 打 (da, 'batting') and 順 (jun, 'order').