Travel from Milan to Castlerotto

This will give you an overview of some options

https://www.rome2rio.com/map/Milan/39040-Kastelruth-South-Tyrol-Italy

There are two Castelrottos in Italy. Bigger one in the Dolomites, tiny one in the Valpolicella wine region near Verona. Which one is it?

I assume the Dolomites one. There are buses from Bolzano it seems, but your life won't be easy without a car. It would be easier in neighbouring Ortisei. Maybe your accomodation host / hotel can advise you better?

Yes, easy peasy. We go to the Val Gardena every year and never rent a car. Train to Bolzano, bus to Castelrotto or beyond as you desire. Buses serve the area very efficiently and of course the lifts take you up to hiking and vistas. Lifts close down in off-season so be aware of that. When are you going?

Here is the schedule for the Bus 170 from Bolzano to Castelrotto. Note German names on the left and Italian on the right (each town has a name in each language). A complete trip planner for this bus line is here. If you are only going to Castelrotto, all you need is the 170.