From Cortina go up to Passo Falzarego and take the cable car to Lagazuoi. The whole top was a fighting area and has even changed shape as Italians, at a time, blew up a whole side mountain. It is even possible to walk back to base station through a long elicoidal tunnel, in the shape of a cork driver, carved in the rock by soldiers to bring up mines (this hike is not difficult, but better undertaken by fit persons with appropriate equipment). When you travel up the cable car, remember that Austrians had the pass and top area, but Italians were clinging to a narrow ledge in the middle of the almost vertical rock wall. Austrians bombed them from above and from the pass for years, without being able to dislodge them.
Another hike, less dramatic, is to the Monte Piana area near Misurina, adapted as an open air museum. It will take a few hours as access is only by foot or by shuttle jeep and the area is somewhat dispersed.