I was just there 1 week ago. If you like wine I would highly recommend visiting the Abbazia di Novacella (abbazianovacella.it), a beautiful medieval abbey about 20 minutes walk (through gorgeous apple orchards) north of Brixen that makes excellent wines, including Sylvaner, Kerner, Gewurstraminer, Pino Grigio, Lagrein, etc. On site is a bar where you can sample the wines, and a wine shop for buying them. Nearby Brixen is also a charming town worth visiting.
I stayed (and ate) at Unterwirt, in the hill-side "village" of Gudon, just above the town of Chiusa - NE of Bolzano. Highlights were a deliciously chewy pasta made with rye flour served with speck (the local cured meat) and local wild mushrooms, and tortelloni of oven-roast tomatoes with parmesan cream and a spinach puree, and parsley soup. The wine list has a section devoted to the Val d'Isarco, the local region within the Alto Adige.
I also enjoyed the outdoor food market and several of the wine bars along the Piazza delle Erbe in Bolzano, but didn't really spend much time in that city.