I would check ahead of time to see if NCL has any type of day care facility to take care of the baby ... at least part of the time. If not, I would seriously consider leaving the baby with relatives at home ... especially if this is your first cruise.
The child must be two years of age to enter the "free" kids crew daycare program. So, you're goinng to have to rely on your relatives for "free" time away from the baby.