Bolzano has no airport operating regular passenger flights.
The closest airports are in Verona and Innsbruck. You might also consider to land in Munich, Milan, Bergamo, Venice, Bologna.
Bolzano has direct train connections to Munich (4h), Innsbruck (2h), Trento (30′), Verona (1h30′), Milan (3h30′), Bologna (3h), Florence (4h), Rome (5h) and Naples (6h). You need a changeover in Verona if you come from Venice (3h). Visit the websites of Trenitalia, Italo, ÖBB, and Deutsche Bahn for your convenience.
A company operates shuttle connections from major airports to Bolzano center. Airports include Milan (Malpensa and Linate), Innsbruck, Munich, Verona, and Bergamo.
You can also choose to travel with the low-cost long-distance bus service offered by Flixbus. Through this service, Bolzano has direct connections with Innsbruck, as well as major Italian and German towns.
Here is also the official website for public transportation in South Tyrol.