We had a less than 24h experience in Switzerland this summer. We took 100CHF out of an ATM and tried to use it up but still had some left over. I don't think we would have taken any cash out, we would have been fine with our no fee CC, but we were at a large outdoor city center music concert with multiple food and drink vendors, most of whom only took cash.
Otherwise we could have purchased our dinner and other snacks with a credit card. We also needed to pay a fee at a parking garage but used a CC for that transaction also.
If you'll be landing in Zurich, taking a train to your overnight Swiss destination (already have train ticket purchased or buy with CC at ticket window), sleep and eat breakfast at Swiss hotel (prepaid?) and then travel on to Italy in the morning I don't think I'd get any Swiss francs. Are you flying out of an Italian airport or returning to Zurich for your flight at the end of your trip?