First of all, your baby is still very young. At less than one month, it is normal that he needs to be fed around the clock. His stomach is still very small, so he can't really stock up on food! Also your baby probably doesn't know the difference between day and night yet.
Sleeping through the night is not really about how the baby is fed or how old is the baby. It's a question of brain development and maturation and it's different for every baby. For example, I breastfeed both of my babies in the same way, but one slept through the night at 2 months and the other at 10 months. They were just different.
I have no magic solution to make your baby sleep through the night (and I'd be rich if I had one!), but at this stage I think that it's important that you help your baby differentiate day and night. For example, you could have a particular routine for the last feed of the day (feed baby in a special chair in his room, dim the lights, have a quiet environment, sing a night time lullaby, etc.). This will help to establish his sleep routine. While feeding during the night, keep the lights to a mimimum, keep baby in the same room where he is sleeping (if possible) and keep interactions to a minimum (do not speak to the baby, sing or play). In the day, it would be a good idea to feed baby in a different setting than at night, for example in the living room with the normal noises of the house and the sun light coming in.
Keep a consistent routine and your baby will eventually sleep in longer stretches. Typically he will have his longer period of sleep right after the last feeding of the day, and will progressively sleep longer and longer (regressions will happen, but they are normal).
Concerning the feeding schedule, I don't know much about bottle feeding but if your baby is gaining weight well and is generally in good shape I would advise to follow your baby's cues. Wait for the baby to show you that he is hungry and let him establish his own schedule. For example, my babies would typically feed more in the evening as they were preparing for the night. Follow your baby and don't forget that his needs will change from time to time as he is growing and developing at his own pace.
Sleep deprivation is HARD, but I'm afraid it's just the nature of caring for a newborn baby. I have the mantra "it's just a phase" and it helped me through many parenting hardships :)