For us this was a pretty big deal because it was the first time Thatkid ever spent the night away from us. Certainly he had been put to bed by babysitters, but we were always there when we woke up.
In addition to that, there's the stress (on us) of arranging for childcare when you don't know exactly when labor is about to begin.
We talked to Thatkid about how when the baby came, I'd have to go to the hospital. He had two possibilities for what was going to happen to him. Option one was grandma coming to stay with him! How fun! Option two was a sleepover with Little HW! At her house! We talked about these two ideas for several weeks to get him used to the idea. Knowing that babies often like to come in the middle of the night, we also warned him about that, and that mommy might be gone when he woke up, but grandma would be here. He asked if we would wake him before we left if that happened. (Side note - we did that, but he basically told us to leave him alone because he was sleeping and had no recollection of that the next morning.)
We asked Thatkid if he wanted to pick out a special present for the baby, and he did.
We also got Thatkid a present (or two) from the baby. I had mixed feelings on this, because Thatkid is suspicious enough to ask how a baby in my belly would get a present, but Toxicesq still remembers the gift my dad got her when he was born, so I decided to go with it. And Thatkid loved it. "How did he know I loved Ironman?"
When the big day arrived we had Thatmom bring Thatkid to the hospital after we were all cleaned up and settled into the room.
This next piece is going to vary by family, but for us, it worked out well to have Thatboy go home that night. He was there to put Thatkid to bed, and it gave Thatkid a sense of normalcy in this time of change. It was a little weird for me to be in the hospital all alone with a tiny baby, but not terrible. And it also guaranteed that Thatboy would get some sleep and be more useful the next day than he was after Thatkid was born.
The next day Thatboy brought Thatkid back for a longer visit. The key to this visit is letting the older child control interactions with the new baby. I didn't force him to play with the baby, nor did I tell him he couldn't interact. Let "no" leave your vocabulary for a couple hours. Which means if your 3 year old wants to do skin-to-skin with the newborn, you let him. (With supervision of course)
I also have fond memories of cuddling the hospital bed with my mom and eating jello after Thatbrother was born. So we did a little of that too. (Although he had graham crackers and cranberry juice instead of jello. And maybe you can guess where the television set was in the room?)
All in all the hospital visit went really well. The next big challenge of course involved bringing the baby home and adjusting to life where he was no longer the center of our universe.