What i can tell you is that since about 1835 passaports were needed by law for any Portuguese leaving Portugal. If your great grandfather left Portugal legally and since he was about 8, most likely he did, somewhere in on one of 18"States" of Portugal his passaport records are kept. Is there in his naturalization papers any word that might be the name of a town in Portugal and you read it as something else, since you stated that "Its hard to read and understand".
If u can / want, send me a scanned copy of the naturalization paper and i will go over it looking for any clue, mainly a name of a town. If one is stated, from it, there is a good change to locate his passaport registration.