DS2 is 4.5 months now, but when he was born I genuinely forgot how bad the sleep can be. For the first 8 weeks he wouldn't sleep anywhere but my chest. Not in his crib, not on DH, solely on me. I was starting to see things in the corner of my eye, shapes etc and was about to call the GP because I thought I had post partum psychosis. DH sent me to sleep in another room one night when DS1 was with grandparents and I woke up a new woman.
Now he's this age, he goes down at 9pm, feeds at 4 and then sleeps until 7:30, which is usually when his big brother comes in to wake us up because he wants to see his baby brother before he leaves for preschool.
The one thing that saved me was DH realising how much I was struggling, so he has taken over the night feeds (I feel a lot less guilty now that there's only one feed in the night instead of three). Mainly because DH will do the feed and fall straight back to sleep, but I'll be wired and up for hours post feed.
DS1 is about to turn four, and he's a dream at sleeping. Quite often, if the baby doesn't wake him up he will sleep until 9:30/10 on a weekend and has done since he was about 18 months.
You cope. It's not great. In fact, it's often hellish but you get through it. I can never nap during the day, when the baby sleeps it's my time to have a cuppa and do some laundry or prep dinner.