This is a fairly recent short story perhaps written within the past ten years. As far as I know it's not by any major author and was posted on reddit or 4chan or something.
A religious group has developed time travel and is sending a man to the past to study their Messiah or some such. The man is preparing for his journey with prayer or reading their holy book.
He is successfully sent to the past, but the rest of his group is not there. He does the work he is assigned alone, spreading stories of his faith. A woman eventually stays with him and he has a family. He develops a following and slowly comes to the realization that the man he came to study is him.
In the end, he does not feel bitterness and is perhaps a little happy. He buries his dog tags or something similar and includes his mission diary for the people of the future to find.