To be realistic here, you have 12 really usable days in Italy. You are likely to be sleep-deprived and jetlagged on your arrival day. You may be the rare person who is functioning at a high level on that day, but most of us struggle. I, myself, am a zombie who has great difficulty simply staying awake. The most I can do is remain vertical and keep putting one foot in front of the other, preferably outdoors. I don't count the first day as a sightseeing day, rather as a day I must somehow survive. And the last day is consumed with packing up, checking out, eating breakfast and getting myself to the airport (usually needs to be three hours before scheduled flight departure), so there's little opportunity to play tourist that day.
Dividing your 12 usable days by 4 means an average of only 3 days per city, and it will be less because of the time lost traveling from place to place, which you can figure will be roughly half a day once you include the checking out/checking in business and traveling to and from train stations. So 2 days in Venice (not counting jetlagged day), 2.5 days in Florence, 2.5 days in Rome, and 2.5 days in Naples. Then you have the final full day when you have to travel back to Rome from Naples to be in position for your flight home; that will allow you a few last hours in either Naples or Rome, depending on how you schedule the train trip.
You'll need to be extremely selective about how you spend your time or you'll feel rushed. I'd recommend only three destinations rather than four.
Some of the major art museums in Florence and Rome are really packed and, frankly, not the easiest places to enjoy under current conditions. Check out the Borghese Gallery in Rome. You have to get your tickets early (slots sell out), but the flow of visitors is controlled via specific entry times, so you are sure of an environment conducive to appreciating the art.
Based on reports of recent visitors, by far the best way to see the Vatican Museums is on an early-morning tour that gets you inside before they open to the general public.
You can save money on your train tickets by buying those tickets early, once your itinerary is absolutely solid. The cheap promo fares are non-refundable/non-exchangeable, so you need to plan first and buy afterward. You can read all about the Italian trains here: Seat61.com. For the trip to Amalfi you'll also need buses or boats.