I really didn't like that version of Bullseye and felt that Kingpin was watered down compared to his normal counterpart.
Kingpin never fought Daredevil or Punisher; he never recovered from the stroke he had brought on by trouble in the family back in 1970. He probably lingered on for a few years before dying.
According to the Marvel Database, that's the Marvel Knights version of The Punisher (which is considered 616):