Can I have a Daidoji Gensai currently at the battlefield use his ability and make a ranged attack even though he didn't move?
Daidoji Gensai movement require to RA?
A topic from AEG's L5R CCG Rules forum, copied for Oracle of the Void from a copy of the forum saved in October 2015, so pages it didn't save are missing. Players' names are left out; the rules officials are named. Rulings are as they stood on these dates; later rulings, the rulebooks and the Rules Archive may differ.
Can I have a Daidoji Gensai currently at the battlefield use his ability and make a ranged attack even though he didn't move?
No. This is why he says "After he moves" - if he didn't move, you don't fulfill the condition, and can't use the Ranged Attack.
Note that since that effect also happens after he moves, if your opponent prevents the movement (via Ring of Earth, Ancestral Armor of the Unicorn, etc) or Interrupts the action to prevent a Ranged Attack (via Turtle's Shell) you don't have to discard the card even if you'd planned to.
On: Daidoji Gensai