This sounds like part of Allen Steele's novel Coyote, which was originally published as short stories. From the Wikipedia description:
Just three months into the journey, something goes terribly wrong. Leslie Gillis, the senior communications officer, is awakened from biostasis. Expecting the year to be 2300, Gillis is horrified when he questions the AI. There was a mix up, and now it is impossible for Gillis to return to his dreamless sleep. His grueling options are either suicide or a lonely existence surviving off the ship’s supplies. While suicide may be more honorable than devouring his crewmates’ rations, Gillis chooses life.
After a brief chat with the AI, Gillis learns that a Mr. Eric Gunther was originally scheduled to awake three months into the trip. Gunther is an agent for the URA, and would attempt to contact the president, or destroy the ship. The program was changed at the last minute to wake someone else instead- Les Gillis. Gillis leaves a note for Captain Lee explaining Gunther’s plans for treason. Why they were changed at the last minute remains a mystery, but it was a change that would cost Gillis his life.
During Leslie Gillis’ solitary life, he did everything he could to keep from going insane, attempting to eat and sleep at regular hours, reading all of the books which were on board, playing chess against the AI, writing stories, and painting. Using practically all of the ship’s art supplies, Gillis created a story about a prince named Rupurt and the fantastic alien world he lived in. He painted scenes of his books on the ship’s inside walls. Eventually, Gillis died in his old age with a fall from the main ladder, after trying to get a better look at the alien ship he had seen. The AI automatically expelled his body into space with the arms of a maintenance bot, and the ship sailed on to 47 Ursae Majoris with no more incidents.
The story was original published in Asimov's as "The Days Between" in March of 2001. Although he does not know the outcome of his action, subsequent stories (or parts of the novel) reveal
his stories are taken up as a part of the mythology of the world, with local animals and various geographical features being named after the ones in Gillis' story.