Not only my first japanese Drama but my first asian Drama was Hana Kimi with Oguri Shun and Ikuta Toma. :)
Great Teacher Onizuka
I don't exactly remember but I think it was Itazura na Kiss, the 1996 version. Then Hana Kimi or it's reversed.
I watched Sprout as my first Japanese drama.
Hammer Session for me~ I found it in my friends laptop, watched it, loved it, and became a jdorama-addict!
i guess it will be beautiful life,,,
Bloody Monday ... I didn't like it and it was hard to end watch it :P
My first jdrama (and asian drama) was Nodame Cantabile. I think it was sort of a spontaneous decision as I really didn't know much about it, or jdramas in general at that time. Shortly after I watched Hana Kimi which was really was got me hooked. 
my boss, my hero was my first j-drama (and first non english show). After that watched my first k-drama (then pretty much nothing but kdrama for a long time). If I did not enjoy my boss, my hero I probably would have never watched another non english show.
Nodame Cantabile for me. The reason I'm shipping jdramas now. ^_^ It's live-action done right.
My first J-Drama was Hana Kimi :D
My first J-drama was Hana yori dango. I think ^^ I was soo obsessed with the manga and drama. But now its like ehhh. I really dont like the relationships in the story anymore... abuse is never okey but I like the actor Matsumoto Jun :)