While you could attempt getting around by walking, cycling, and bus in some parts of Ann Arbor (technically a half hour drive west of Detroit but still an extension of the Detroit Metro area), for the vast majority of the area, a car isn't optional for an easy life.
The Detroit metro area is huge, making public transit slow. Additionally, local governments are resource constrained, so transit service quality is poor. Even getting to the airport is difficult by public transit, much harder compared to Chicago, St Louis.
If you're content in inner Ann Arbor or maybe Royal Oak, you could try car sharing services with public transit but you will be restricted in your shopping, socializing, and work (if away from home).
Be sure to look into car insurance and registration costs if you do end up getting a car.