The order below can shift with your start date, so treat it as the shape of the week rather than a timetable. Each day ends back in the same Naples hotel.
Day 1: Arrival, and the Center on Foot
Nothing is scheduled beyond check-in. Use the afternoon on the Centro Storico, where it is easy to lose the map early and stop caring.
Day 2: The Coast Road Through Sorrento, Positano and Amalfi
Three towns in one day sounds like a rush, and the drive is the reason it works: the road itself is the sight. The day starts at a limoncello factory, then gives you an hour in Sorrento to walk the town, then moves to Positano for the streets or a drink by the sea.
Amalfi comes after that, and it is the stop with something built to see: the Cathedral of St. Andrew. If the afternoon is too hot for it, the shore and a gelato are a fair exchange. The coastline you have been driving is a UNESCO World Heritage site, which is the sort of label that usually oversells a place and here does not.

Day 3: Capri by Hydrofoil
The hydrofoil leaves from the city port and lands you on Capri for the day. From the harbor the day is free time in Anacapri, the quieter half of the island, where the Church of San Michele carries an extra charge and depends on whether it is open. Then the Piazzetta, the small square that functions as the island's living room, and the Gardens of Augustus for the view back at the Faraglioni rocks.
Two warnings worth having. The sea decides the schedule, and when it turns rough the runs are stopped, so keep the evening loose. If you are sensitive to the rolling of a boat, or traveling with young children, take something before you board rather than after.

Day 4: Pompeii, Half a Day in a Roman City
This is the morning the trip is built around for a lot of people, and the format is a half-day excursion with a skip-the-line ticket and a local guide. The guided walk runs about two hours. In that time you cover cobbled streets, villas, temples and shops, and the plaster casts of the victims, made by pouring plaster into the cavities their bodies left in the compacted ash.
The route changes from day to day with site openings and visitor numbers, so no two groups see the same buildings. What the operator guarantees is one from each category: a temple, a market, an ancient shop, a villa, a thermal bath, a theater, and the Forum. Ask the guide that morning which extra buildings are open.

Day 5: Breakfast, and the Road Home
The trip ends after breakfast, with nothing scheduled. If your flight leaves in the afternoon, the morning covers Piazza del Plebiscito in one loop: the Royal Palace on its east side, the church of San Francesco di Paola on its west, and the colonnades running along both.