In Mike Ford's amazing Klingon novel,
The Final Reflection (which pre-dates TNG's version of the Klingons), he came up with Klingons ageing rapidly, but with them continuing to do so and dying young too. I recall a scene where Kirk meets Kor or Kang or one of his old enemies in the movie era (in
How Much For Just the Planet? maybe?), and they're now much older than Kirk.
Maybe someone read that and used it as an excuse to age Alexander rapidly, then someone else decidied Klingons live centuries as an excuse to get the TOS Klingon trio in to DS9.