When getting on the Green Car of a limited express, you have to buy a basic ticket, a limited express ticket for a reserved seat (this ticket costs the same as the limited express ticket for a non-reserved seat on the Shinkansen) and also, a Green ticket.
特急列車のグリーン車を利用の場合は乗車券・指定席特急券（この場合の料金は自由席特急券と同額）・グリーン券が必要。