When starting the game, scroll up and down on either a save slot or a no save slot until you see Tails by himself. As long as you don't get hurt, you'll easily score 100 rings in either Level 1-1 or 1-2. For best results, try to avoid enemies and make sure to grab a shield whenever you can. Also, go through levels slowly and explore what you can. The achievement will unlock once you hear the 1up sound coming from ring 100.