Italy to Austria or Switzerland

Where are you staying in the Dolomites?

To be honest, there won't be a big cultural shock between Italian Südtirol and Austrian Tirol, unless you're staying in the more Italian parts of the province.

I would consider Innsbruck as a bad-weather day-trip option. Otherwise, I think you'll be better off exploring the Dolomites.

If you're west of Meran or somewhere around Madonna di Campiglio, you could consider a day trip to the Engadine region of Switzerland. It's beautiful.

Cultural differences in Alpine areas are very subtle. For example, the Trento area (italian speaking) is very close to the Tirolean area (German speaking); differences between Italian Tirol, Austrian Tirol and southern Bavaria are hairsplitting. German Switzerland is also similar, but speaks a dialect barely understandable to other Germans.
Ortisei is a formerly Latin area (giving origin to the group still speaking Ladin), on which German immigration was superimposed from 15th century and then Italian government in 1918.
Or, to make another example, in the same Swiss valley, some parishes may be Catholic and some Protestant, according to the status of the parish priest at some time in 16th century when a settlement was signed - but no other difference can be spotted in the sparse population. In another case, I have observed how moving from Como, Italy, to Disentis, Switzerland, the local dialects form a sort of continuum from the Milanese dialect to Rumantsch language. In some places, you cannot tell for sure what language the locals are speaking. Spotting cultural differences requires some study and some observation.

If you plan to drive into Austria make sure your car has the required vignette sticker
Also make sure the rental agency allows the car to cross into Austria

I don't recommend any one-day/night trip to either Switzerland or Austria, unless you want to spend all day on the commute. The biggest cultural difference would be in the form of train system efficiency -- Swiss trains are more punctual than Austrian trains, which in turn, are more punctual than Italian trains.

Thank you all for your responses. This is very helpful.
What about the Bernina Express? Are the views impressive? What about staying in a town along the route? Would you recommend taking a train on this route?

A few years ago a friend and I took the Bernina Express to Chur, Switzerland--it was wonderful. We arrived in early evening (September, already dusk) and as we were walking to our hotel we came across a group on the street corner playing long horns--what a lovely welcome. Stayed overnight and took the return train in the morning wishing we had planned to stay longer in Chur. There were many picturesque points along the route. I fact I am considering doing it again this year.

Thank you for all your feedback!
Any thoughts on going into Slovenia instead of Switzerland?