Manolin's relationship to Santiago and his attitude regarding reputation may be the most direct commentaries in the text with bearing on your question.
Early on, Manolin tells Santiago that he will go in Santiago's boat again. He wants to stay with Santiago because he believes that Santiago is a great man and a great fisherman, despite the nearly three month term when Santiago catches no fish. Reputation and luck have no effect on Manolin's loyalty to Santiago.
Santiago refuses to let Manolin come in his boat because Manolin's parents have clearly mandated this decision.
They believe that Santiago's bad luck is real. By extension, we can infer that they believe in the importance of reputation. In their conversation, Manolin and Santiago both state their opinions on this subject, finding no profound importance in the idea of bad luck or reputation and also maintaining a strong hope for Santiago to break his bad streak.
Santiago's poverty serves as adequate explanation for this shared desire to see the streak end. (Catching a fish will bring in money.)
He has little to eat, and frequently must rely on the boy or others in the village to bring him food and clothing.
Again, at the end of the story, Manolin demonstrates his attitude toward reputation when he goes against his parents' wishes and plans to go fishing with Santiago in the future.
Additionally, Santiago is not truly aware of the awe he has inspired with the carcass of the great fish he has brought in. He is too exhausted to get out of bed and has no idea what kind of "reputational effect" may or may not be come as a result of the presence of the great fish lashed to his boat.
The fishermen in the village marvel at the mutilated fish; at eighteen feet, it is the largest marlin they have ever seen. The boy brings the old man food and fresh clothes and watches over him as he sleeps.
Santiago is not completely divorced from the idea of reputation, however, as we see in the passage relating the story of his arm-wrestling success. He is glad to recall how he got his nickname and how he defeated the larger man. This story can be seen as a counter-point to Santiago's humility, which is certainly the more pronounced trait in his character.
As an old man, his self-belief is strong but not attached or derived from any public pride or outward bearing.