There isn't any control lag. The only thing regarding to lag would be characters suddenly popping up right next to you. It never interferes with gameplay like Final Fantasy CC Echoes of Time. The framerate never even drops. I think Mario Kart DS had similar popups.
But yes, the faster your internet connection, the less problems.
And this game is hyped because it's a successor to one of the first online RPG's ever and follows the same style- Phantasy Star Online. Still kicks ass to this day, and the DS game is quite similar.