What are the examples of man versus themselves in The Open Boat?

Based on the wording of your question, I'm not sure if you mean to inquire about character vs. self conflicts or character vs. character conflicts. In terms of character vs. self, I think we see such a conflict within the injured captain. On the lifeboat, he finds himself

buried in that profound dejection and indifference which comes, temporarily at least, to even the bravest and most enduring when, willy nilly, the firm fails, the army loses, the ship goes down. The mind of the master of a vessel is rooted deep in the timbers of her, though he command for a day or a decade [...].

The captain seems to struggle against this dejection and indifference, continuing to issue instructions to the oiler and others in order to maintain the proper direction; however, he also seems to give into these profoundly upsetting feelings from time to time as well.
In terms of character vs. character, we see such a conflict between the cook and the correspondent when it comes to life-saving stations and houses of refuge. The cook is certain that a certain house of refuge is nearby and that they will be spotted by the crew. However, the correspondent argues, saying, "'Houses of refuge don't have crews.'" The two bicker back and forth for a short while longer.
Despite these minor conflicts, the major conflict of the story is the one of character vs. Nature. It is nature against which the men struggle: the waves, the sharks, the sun, the saltwater (as opposed to fresh, potable water). It is nature that does not care whether the men live or die but just keeps doing the natural things that nature does.