It is true that many COs will try to time their periods of sleep for when the operational tempo is lower as a risk management measure. I can only assume the Fitzgerald's CO had made his risk assessment and figured that it was better to sleep at night than to sleep during the day when the contact density would be even higher.
But for a Navy warship being "approached by other ships" is a fact of life for every hour of every day, you can't always plan your sleep schedule for when you're the only boat on the pond.
But this still relies on the watch team detecting the contact and realizing that it will pass near to the ship, and so again the ship's captain is back to relying on his or her watch team to at least get to that minimal level of competence -- because if they can't rely on the officers on watch to be able to wake them up properly, we're back to captains who can never sleep.