I havn't been to Rincon in years but one thing I know is it will probably cost you 2 or 3 days of the cost of a rental for a cab each way. I would consider the cost of the rental car as part of the trip.
We have stayed in Rincon 3 or 4 times. I could not imagine staying there without a rental car. There is a small downtown area with some restaurants, but I do not remember it being close to the beach. Rincon was very friendly and laid back. Enjoy!