Which floor should be done first when updating a multi-family residence?
I don't think there's a good answer to this because plumbing needs a full stack, from the drain in the ground, to the vent in the roof. When you move utilities from one location to another, if you haven't changed the room above, you're stuck with lines on both new and old locations in the floors you've renovated until you're done with the renovation. And if you try to move utilities without first renovating the rooms below, you either have to open the rooms up below or run connections from the new back to the old location. In the end, I fear that you're going to spend more money trying to do it one part at a time rather than saving up until later and doing it all at once.

To partially answer your question, when we are building a new home, we construct from the ground up, for structure, and it's also the source of utility lines so they go the same way. When we finish the build, we go from the top down. This way we aren't tracking dirt over finished carpeting and scratching up freshly painted walls.