I really enjoyed Orvieto and think you could keep yourself busy for two days (for one thing, you could spend hours just wandering the picturesque streets), but there are many wonderful places in Italy, and one day would allow you to see a lot. These are the things I remember:
Cathedral (don't skip the pay-to-enter chapel)
Cathedral museum
Etruscan museum
Archaeological museum (might skip if you're doing the big one in Naples)
Walking tour (check with tourist office)
As of 2015 there was a branch of the T.O. near the funicular you take to the upper town. I think it was on the lower level near the train station. You can pick up a map there, ask about tours, and see whether you think the city card would pay off for you (I'm guessing not on such a short stay). It includes a round-trip on the funicular and a lot of local activities.
I stayed at the Hotel Corso, a moderately priced place that is nonetheless at least one cut above my usual. I'm sure there are fancier places in town. It's closer to the top of the funicular than most (perhaps all) of the other hotels, which makes it a bit of a walk (albeit an attractive one) to the core of the historic district. I had a too-heave suitcase and was greatful not to have to walk all the way (somewhat uphill) to the area around the Duomo. Near the Corso there's a gelato stand about a block back toward the funicular, and there's a small fruit/vegetable stand directly across the street.
I didn't take breakfast at the hotel; there is an extra charge. But I did peek into the breakfast room and thought the spread looked decent. I was there in early June, and there was a plate of melon. Definitely not just coffee and a few bread products.
Orvieto has a lot of good restaurants. My favorite was Cibus, which is Sicilian and focuses on seafood. It's worth spending some time online looking for recommendations, because I assume there are also a few places to be avoided.
I was in Orvieto for 2 weeks taking Italian classes, so museum closing schedules were not critical to me. Since you'll only have one day there, it would be prudent to check hours of operation for the fall.