There is no reason to believe that this novella, The Old Man and the Sea, is written in anything but chronological order. It begins with these lines:
He was an old man who fished alone in a skiff in the Gulf Stream and he had gone eighty-four days now without taking a fish. In the first forty days a boy had been with him. But after forty days without a fish the boy's parents had told him that the old man was now definitely and finally salao, which is the worst form of unlucky....
From this inauspicious beginning, we get all the antecedent action (that which happened before our story begins) as well as a hint about the future, what is to come. We know the old man, Santiago, is now marked as having bad luck.
The story takes the old man to sea, has him wrestle with his brother the fish, and brings him back home. There is no evidence that the story moves either back or forward in time outside of a normal chronology. Now, you do qualify your question with the idea of flashbacks, and the novella does contain those, to a degree. We experience Santiago's dreams (Africa and the lions) as well as his reminiscences (of Joe DiMaggio and baseball, as well as his wife). These wanderings of the mind which Santiago experiences throughout the course of the novel in no way change the forward movement or the chronology of the story, though. They are simply a looking back by an old man who is tired and thinking and dreaming about other things to help him survive his present.