Packing a New York apartment is a space problem before it's a time problem. There's rarely a spare room to fill with boxes, so the trick is to pack in waves: start with what you won't touch for two weeks, and keep a clear path through the apartment the whole time.
Begin with closets, books, and anything under the bed. Then do decor, extra linens, and off-season clothes. Leave the kitchen and bathroom for the last few days, and pack a first-night bag with chargers, toiletries, medication, and a change of clothes.
Use small boxes for heavy things and large boxes for light ones. A large box full of books is too heavy to carry safely up a walk-up. Label every box on two sides with the room it's going to, not the room it came from, so it lands in the right place at the new apartment.
If you'd rather not build boxes at all, reusable plastic bins stack neatly in a small space and don't collapse, and a professional packing crew can do an entire one-bedroom in a few hours. Either way, the goal is the same: on moving day, everything is packed, labeled, and ready to go out the door.