Any mode will work with this: Outrun, Heart Attack, or Time Attack. You have to finish for it to count. Running out of time or restarting will not count as a game. I got this after finishing a 15-course time attack and it popped when asking for my name for the score board. So make sure it at least gets to the title screen and saves for it to count. This does not have to be in one sitting.
The quickest route if you just want to complete games is goal D (left, then all right). Unless you are really good, you will easily play 50 games when trying for the other single player achievements. It probably took me 100+ complete games minimum.