Is one day visit to Naples worthy ?

If you have your heart set on visiting Naples, I'd suggest doing it as a day trip from Rome rather than relocating for 1 night. The high speed trains get you from Roma Termini to Napoli Centrale in just over an hour, and you may be able to get tickets for as little as 20 euro each way by booking in advance versus paying 46 euro each way on your day of travel. Both Trenitalia and Italo high-speed trains operate between Rome and Naples. Check schedules and prices for both before buying, and keep in mind that the least expensive tickets are non-refundable and non-exchangeable.

Maybe tell us what sites you want to see so we can tell you if it's feasible.

Yes it is worth a couple of hours; you should familiarize yourself with what is there and decide what you want to see.

It sounds like you have 14 full days on the ground and 4 destinations you'd like to see that have some minor travel time between them. Naples is definitely worth a day (it's worth more than that actually), but I would save it for another trip that's focused on Southern Italy instead of trying to rope it in on this trip. This is just me personally, but I would be frustrated if I went somewhere for a very quick stint and ended up shortchanging the place (in other words, it seems too rushed in the context of everything else you have going on). Naples is a big city, it really does deserve more than a day (and that doesn't even count Pompeii).

After reading your comments I think we will leave it for a day trip if we feel like while we are in Rome. We would like to come back some day to visit the South, so it will be definitely in our list for the next trip.
Thanks a lot for your comments.

Naples is worthy but I think you should save it to combine with Pompeii, Capri, Amalfi Coast on another trip.

We are not sure if it is a good idea to travel to Napel on May 14 for
one day, departing the following date to Florence, or just skip the
city.

With the very little amount of time that you have to give it, I would skip it. Save Naples for a future 'South of Rome' trip where you can combine it with Pompeii, Herculaneum, the Amalfi Coast and other locations in that general region. It'll save you a one-night jump, and give you a spare night for another interesting location that doesn't involve time-consuming backtracking, or which is a great landing spot for day trips, such as Florence.

I would spend the 14th in Rome and take an evening train to Florence, rather than go to Naples (which I quite liked). You won't see much of the city on the 10th, by the time you get through the airport and to your hotel. You may be just trying to stay awake until it's late enough to go to bed. Even the 11th, you could still be in a haze in this large, bustling city. That gives you 2 full days to appreciate the major sights. Use the 14th to see more of the sights, have a picnic dinner on the train, or go out for a latish dinner in Florence.

Milan? Okay, I understand that everyone has different tastes, but the general opinions I've seen here are that Milan is not worth 3 nights and since it's not in a "straight line" from Florence to Venice, I'd go somewhere else. Bologna is a good place for 3 nights. You would go through Bologna to get to Venice from Florence and since Bologna is a major rail hub, it's very easy to day trip - Ravenna (far, but worth it), Ferrara, Modena, Padua to name a few.