Hike the natural park with waterfalls just north of the Cortina city center. There's also exciting "Via Ferrata" routes to traverse the ledges if you have the safety equipment to hook into the cables and some training.
Having a car in the Dolomites is very handy. There are public buses but they only go to the most popular places. That would work with a short trip with one local hotel in let's say, Cortina d'Ampezzo, Castelrotto, or Ortinsei.