What is the ruling on performing religious duties, such as Hajj, for a person who has been legitimately imprisoned? Is he excused or does he bear sin for being the cause of his imprisonment? Or should the ruler release him to perform Hajj and then he completes his sentence?

Hajj is not obligatory for a prisoner while incarcerated. Rather, it becomes obligatory upon the removal of the impediment, which is release from prison. A prisoner is prevented from leaving to perform the obligatory Hajj because he anticipates being able to perform Hajj himself at a later time.