Final answer:
The Golgi apparatus is the organelle responsible for processing and shipping proteins. Proteins synthesized by ribosomes on the rough ER are sent to the Golgi apparatus for modification and packaging.
Step-by-step explanation:
The organelle that processes and ships proteins specified by the nucleus is the Golgi apparatus. This is part of the endomembrane system which includes the nuclear envelope, lysosomes, vesicles, endoplasmic reticulum (ER), and the plasma membrane. The sequence starts in the nucleus, where DNA is transcribed to make RNA. This RNA exits the nucleus and enters the cytoplasm where ribosomes, particularly those on the rough ER, translate the RNA into proteins. These proteins are then sent to the Golgi apparatus where they are modified, tagged and packaged before being transported to their destination within or outside of the cell.