Yes, there's plenty of locales he can go in casual clothes. The only places on NCL ships with any dress requirements, are the restaurants during "dinner". And that is minimal, RESORT CASUAL. For a man, that means slacks and a collared shirt. A polo shirt, an Aloha shirt, a dress shirt all qualifies - as long as it has a collar. Slacks don't have to be dress slacks either, Docker style slacks, kahki slacks are good enough. T-shirts and athletic shirts aren't allowed in the restaurants during dinner.
Jeans are questionable, offically they aren't allowed in the restaurants during dinner, but unoffically many get away wearing clean jeans. But the 24 hour food court restaurant and the buffet offically allow jeans and shorts during dinner. For breakfast and lunch, jeans are allowed at all the restaurants which are open.
Looks like NCL and your husband are made for one another.