Final answer:
The Golgi apparatus is the organelle responsible for the cell's packaging, storage, and shipping, acting like a cellular post office.
Step-by-step explanation:
The organelle that controls a cell's packaging, storage, and shipping functions is the Golgi apparatus. Often likened to a post office within the cell, the Golgi apparatus receives proteins from the rough endoplasmic reticulum (ER), then sorts, modifies, packages, and tags these proteins before sending them off to their destination. It produces new organelles called lysosomes and repackages vesicles that then may either be transported to other areas of the cell or exported out of the cell through exocytosis.