Brixen / Bressanone is a popular ski resort town in the northern Dolomites about 30 kilometers from the Austrian border. It's also the start of the Alta Via 2, one of the world's most famous long distance alpine treks. It's not famous for its mountain bike trails but there are some good ones there.
The best of them start at the top of the Plose goldola. There is a restaurant there along with a bike hire place and map boards.
You can ride the four wheel drive tracks all over the ski fields and take in the stunning views of the Dolomites to the east and the local mountains elsewhere. You can take a break and refuel with great food and coffee at the Refugio Bressanone near the top of Monte Telegrafo.
Best ride of all the the down-hill flow trail that goes all the way down under the gondola. The surface is juddery though so you need to hire a good bike if you really want to flow! You can leave your hire bike at the bottom of the gondola.