If your baby is breast-fed, only you can supply what she wants at 4 a.m. Thankfully, a breast pump can make your nights run a bit more smoothly.
You and your husband should take turns feeding the baby when he wakes up in the middle of the night needing nourishment. Keep a couple prepared bottles in the fridge, and teach your husband how to ensure the bottles are at the correct feeding temperature.
Then let him take the next 4 a.m. call. He should be willing to, because he loves you and appreciates your need for rest. Let him prove his dedication to his child by taking over a few of those night-owl feeding times.