From main menu, go to Options.
Change difficulty to Easy and turn Supertars mode OFF.
Progress through Singles or Doubles tournament from main menu.
Play with Ai Ai.
When you make it to Dr Eggy, return the ball to the opposite side of him with X, A for a low shot just over the net and to either side.
After he returns this shot, nail it with A and a hard angle(<-- or -->) away from him.
This achievement was giving me problems, but I got it 1st try using this method.
You will find this achievement MUCH easier to unlock if you follow this.
You will need at least an AA rank for each match in order to face Dr. Eggman.
This is accomplished by winning each match without losing a round.
If you must leave a negative, have some guts and leave a comment explaining why.