I am making my way through Jeanie's Kdrama classics list (& also some Japanese ones) & they are well worth the effort!!
I only count 2005 and before as old, so my favorite "oldies" are:

Dramas

(1998) Princess Pearl
(1998) Great Teacher Onizuka
(2003) Kimi Wa Petto
(2004) Orange Days
(2004) It Started With a Kiss
(2005) Hana Yori Dango
(2005) Devil Beside You
(2005) 1 Litre of Tears
(2005) My Girl
(2005) Prince Turns To Frog

Movies

(1987) A Chinese Ghost Story
(1999) The Road Home
(2003) Memories of Murder
(2004) New Police Story
(2004) House of Flying Daggers
(2005) Marathon
The "oldest" dramas I've watched are from 2000, such as Summer Snow, Ikebukuro Westgate Park, Trick and Beautiful Life, but my favorite from this time is definitely Yasha, it still ranks in my Top 5-dramas
The oldest drama I've watched is Virgin Road (Japanese, 1997) and I loved it! :) It's the only drama from the 90s that I've watched. Other older favourites: Orange Days (2004) Kimi wa petto (2003) Koukou kyoushi 2003 (2003) Itoshi kimi e (2004) Gokusen (2002) Slow Dance (2005) Antique (2001)