Having your first baby is an enormous shock to the system. If you're not used to babies, you have no idea what life with one is really like. And if you've never had to get up several times a night, you have no expectations about what the tiredness feels like. It's very very hard.
The most important thing I learned is that you do whatever it takes to get some sleep! Your baby is still tiny and, unfortunately for their tired mummies, many of them just want to be snuggled close to you. You said that he sleeps well during the day. Do you nap when he does? It can make a huge difference. With DD1 I still used to lie down sometimes when she was 2! Also, while you have your DH at home let him take the baby sometimes when he's settled so that you can go and get some sleep. Will you have any family or friends who can come over sometimes when he's gone back to work?
It's a good idea to keep night feedings quiet and dark. Some babies, like my DD1, start to feed less and sleep well at night early on. Others, like my DD2, feed reguarly during the night for a long time, so you have to decide what will work for you and help you to get some sleep. With DD2 I had to co-sleep for survival as she fed so frequently at night. I had never intended to do this but if the lack of sleep is impossible for you, this may be the only solution (whenever she woke, I would switch her over to the other boob and go back to sleep - I actually woke in the morning feeling quite refreshed!) . There are ways to make it as safe as possible: no pillows or bedding near baby. It also helps if DH sleeps elsewhere. Or you could try one of those side-car cots that attach to the bed so that baby is close to you but has his own space?
Things will definitely improve, but there is no set in stone date to say when unfortunately. Good luck, you will honestly look back on this time fondly one day!