Once born, how did the alien grow so quickly?
Just as it is likely the alien did not grow inside Kane using Kane’s nutrients, it is also unlikely that this alien could grow using organic food of the sort the crew was eating. The alien is a bio-mechanical creature, its blood is acidic, and its exoskeliton appears to be very solid. A possible means of such rapid growth is for the alien to spray an acid on a metallic surface and ingest the metallic substance that remains, building its body mass with it. Since this is inorganic growth, we are not left with the unlikelyhood of the alien growing so rapidly with mere food.
The Xenomorph (Alien) is NOT biomechanical in any sense of the word; it is a biological life form; it not only uses humans for the gestation of eggs, but also for food.
The reason it grew so quickly is due to its quick metabolism; the real question is, what did it eat?
Simple but most likely answer: mice or rats, and possibly insects. We know there was plenty of water on board, and it’s likely the stores on the Nostromo included extra food in case there was an issue with the hypersleep chambers (food was also likely stored on the refinery section in case there were cases with famine on some outer colonies).
Mice or rats would definitely been on the ship (which makes the tabby cat Jonesey not only a pet, but a working feline, keeping the rodents away from the sections of the Nostromo where the humans would hang out in once woken from their cold slumber.
With plenty of rodents to chow down on, and nearly unlimited water to drink (remember the scene where Harry Dean Stanton enjoys the water dripping on his face in one of the storage areas?) , the Xenomorph would grow in a day or two to full size – time enough to get to the size where he could eat the brain of the actor who played the character "Brain" from 1981’s Escape From New York.
