One after the other or one after another means taking each one individually following a sequential order. In this case the author provides a a list of occupations: first he was abbot, then officer, then scholar, and so forth.
Among does not imply any order, just one in a grouping, and so is not equivalent here. One could write
He was, among other things, an abbot, an officer, a scholar, a writer, a banker, a con artist, a magician, an infantryman, a spy, and a diplomat.
and be factually correct but less informative. By this phrasing he could have been a magician first, then later a banker, and then an abbot, and so on.