As the Hemingway code hero, Santiago does not feel despair; he continues bravely. However, he does know that he is defeated:
He knew he was beaten now finally and without remedy and he went back to the stern and found the jagged end of the tiller would fit in the slot of the rudder well enough for him to steer.
If he were in despair, Santiago would not have acted immediately on steering his boat. Furthermore, after the last shark goes, the old man feels the coppery taste in his mouth and becomes afraid of it temporarily. But, he spits into the ocean defiantly and tells the shark,
"Eat that, galanos. And make a dream you've killed a man."
As further support that Santiago is not despairing, but endures, at the conclusion of the novella, he may be destroyed in body, but he is not in mind, for he "was dreaming about the lions," a dream that is symbolic of power and resistance.
On page 119, of Ernest Hemingway's The Old Man and the Sea, Santiago has found success in bringing in a great marlin. Santiago's pride soon disappears when sharks surround his fishing boat and eat the marlin right off the side. Although Santiago never gave up fighting the sharks, they were successful at taking away Santiago's prized fish.
After returning home, which takes him a very long time, Santiago collapses upon his bed. Manolin comes to see Santiago the following morning. Santiago seems broken and filled with despair. This despair comes from Santiago's thought that he had been defeated. He did not bring in the marlin as he had wanted to do.
Santiago, initially, does feel like he lost hope. Given his struggle with the marlin and the sharks, he feels as if he has nothing to show for his efforts. His initial reaction, to the loss of the marlin, is one of despair.
I hope that the edition I am using is close to the one you are. The answer comes from Day Four and Day Five.