When Apex Altisaur enters the battlefield, its first ability triggers.
Holding priority, activate Gyome twice by paying ({2} magic symbol) and sacrificing two Foods, giving Apex Altisaur and any creature an opponent controls indestructible until end of turn.
Resolve the Apex Altisaur trigger, causing Apex Altisaur to fight the indestructible creature the opponent controls.
Apex Altisaur's second ability triggers, and Fall of Cair Andros triggers if the opposing creature's toughness is less than 0.
Resolve the Fall of Cair Andros trigger, if applicable, causing you to put a number of +1/+1 counters on an Army you control equal to the excess damage.
Resolve the Apex Altisaur trigger, causing it to fight the indestructible creature the opponent controls.