Monstrous isn’t an ability that the creature has. It’s just something true about that creature. If the creature stops being a creature or loses its abilities, it continues to be monstrous.

Once a creature becomes monstrous, it can’t become monstrous again. If Kalemne’s Captain is already monstrous, the activated ability won’t do anything.

An ability that triggers when a creature becomes monstrous won’t trigger if that creature isn’t on the battlefield when its monstrosity ability resolves.