I’m no expert, but I’ve looked at this in the past. Most large buildings have their own water tanks inside them. If you think of an image of a New York skyline away from the skyscrapers you might picture small water towers on top of most of the buildings. Those are the tanks that supply water for the building. Skyscrapers will have multiple tanks inside the building itself, maybe one every 5 or 10 floors up. There will always be one at the top of the building but it might not be as obvious looking as a small water tower. Each tank will serve just the floors in between it and the next tank below. When the tank needs to be refilled it just draws from the tank below it. This way the building doesn’t pump all the water it needs all the way to the top floor; it only pumps water as high as it needs to go. Keeping water in a tank means it still works like a traditional gravity-fed system and should function for a while even in the event of a power outage.

Handling wastewater is relatively easier, it still just needs to flow down. The pipes just drain into one or more pipes going down to the bottom of the building.