I've never found anything reasonable to eat on Cap Ferrat, but I've never even tried. Virtually all the little spit of land is residential. There are one or two shops at the very bottom of the bus route but I've never looked for food there.
Nearest, to the north, is the small town of Beaulieu-sur Mer which has some nice shops, a few coffee shops and restaurants, a decent supermarket (U-Marché), and a decent cheap(ish) hotel with a great view. It also has wide and large parking spaces and reasonably wide road.
Just halfway between Nice and Cap Ferrat, and walking distance from both is Villefranche-sur-Mer which is much bigger, has lots and lots of restaurants and shops, is very busy during the day if the cruise ships are anchoring (and tendering in, because they are too big for the port). There are restaurants up in the town, and down by the docks and harbour. A note, if you are still in that car, as wide as the streets are in Beaulieu-sur-Mer they are extremely narrow and winding in Villefranche-sur-Mer. Both Beaulieu-sur-Mer and Villefranche-sur-Mer have easy to find train stations and bus routes.
Villefranche-sur-Mer is prominent in Rick Steves' Riviera and Provence book and lots of places to eat are there.