Small premise: if you stay in the historic center of Naples (Via dei Tribunali, Via San Biagio dei Librai, Via Duomo, adjacent and nearby streets) you can safely convert the day 2 to day 1. In this way, also according to your arrival time in the city, It will be easier and more convenient for you to visit.
Day 1: Start your city tour from Via Toledo (I recommend you to arrive by metro so as not to miss what has been considered by "The Daily Telegraph" and CNN the most beautiful metro station in Europe and the world; also, when you reach the escalators, keep your eyes high and your phone at hand, I assure you that you will not regret it!)
Via Toledo (built in 1536) is one of the main arteries of the city, vibrant street, full of shops where to stop for shopping, souvenir items, artisans, street artists, shops where to taste the classic street food of Naples (pizza a portafoglio, cuoppo fritto, fried pizza, various desserts such as babà and hot sfogliatelle). You will have to walk in the direction of Piazza del Plebiscito.
On the right of Via Toledo develop the Spanish Quarters (they are so called because during the period of Spanish domination, we talk about 1500, the military troops settled on the hill that today hosts the Quarters. The position was ideal, first because close to the Royal Palace, and then because, thanks to the slope of the hill, the soldiers managed to quell popular uprisings by pouring boiling oil on the streets), here you will find a world apart, a city in the city, small streets that intersect, restaurants and pizzerias, historic buildings and walls that speak thanks to their murals. The most iconic is at the end of Via Emanuele De Deo, in largo Maradona, where we find the mural dedicated to Diego Armando Maradona, for many Neapolitan a god. Place of worship, where people from all over the world come to pray, take pictures, buy gadgets or simply to snoop.
After visiting the Spanish Quarters, return to Via Toledo and head towards the Galleria Umberto I.
The Galleria Umberto I is a majestic shopping gallery built between 1887 and 1890 and dedicated to the then King of Italy. Within the gallery you will find cafes, bookstores, shopping shops. (Here too, I recommend looking up to admire the beauty of the dome of the gallery and the details in baroque style).
Exit the Gallery in direction of Via San Carlo (here you can also admire the main facade of the Gallery) and continue left towards Castel Nuovo (for the Neapolitans the Maschio Angioino).
The Maschio Angioino is a medieval and Renaissance castle, built between the thirteenth and fifteenth century in Gothic style. Symbol of the city, dominates the scenic Piazza Municipio, seat of the municipality of Naples. Inside the castle, paying a ticket of 6 euros, you can visit the interior rooms such as the Hall of the Barons, the main one of the Castle, frescoed by Giotto, with the splendid octagonal vault and the Hall of the Armory where archaeological remains from the Roman era have been found. The Palatine Chapel is the only evidence of the ancient Angevin palace and there are some sculptures of the Neapolitan Renaissance. The Castle houses the Civic Museum with frescoes and paintings ranging from the fifteenth to the eighteenth century and works of art from the nineteenth and twentieth centuries. You can also visit the prisons, where a legend is hovering: here prisoners were locked up and mysteriously disappeared. After careful checks it was discovered that a crocodile dragged the prisoners into the sea to devour them. From then on the crocodile became the executioner of all the prisoners of the castle!
After leaving the castle, return to the Umberto I Gallery and go to the nearby San Carlo theatre.
The Teatro San Carlo is the opera house of Naples, built in 1737, and it is the oldest opera house in the world to be still active. For 9 euros, you can admire its immense hall, its horseshoe structure, the oldest in the world, inspiration for all the theatres of Europe and listen to the stories of the theatre and all the great artists who have performed there over the centuries.
Leaving the theater will be easy to reach the immense Piazza del Plebiscito, the largest square in the city and one of the most important in Italy. Symbol of the city, it is a must to stop here to enjoy its beauty and majesty, dominated by the Basilica San Francesco di Paola and the royal palace. Here you can enjoy a good coffee at the famous café Gambrinus, literary café built in 1860 and which has hosted illustrious figures such as Ernest Hemingway and Oscar Wilde.
As mentioned in Piazza del Plebiscito is the royal palace of Naples, visiting it will be like going back in time!
With a full ticket of 15 euros (free under 18), you will find yourself catapulted into the 1600s. The part I prefer is just the beginning, or the staircase of honor that dominates the entrance to the palace. With its grandeur, its white marble, the statues, the antique pavement, you will feel like a king or queen of the Renaissance. The visit continues by visiting the various rooms of the palace, with touches of extreme beauty to the court theater, throne room and royal chapel. You can also visit the hanging gardens with a view of Vesuvius and the time gallery at certain times with the full ticket. After visiting the inside of the palace you can also enjoy the outside with its wonderful gardens, called romantic gardens.
After the visit, you can continue the day in the direction of Via Chiaia, Piazza dei Martiri and Via dei Mille, famous streets for strolling and shopping where you can also find great clothing brands and not only (in Via dei Mille) like Gucci, Louis Vuitton or Rolex.
The day can only end with a walk on the most beautiful promenade in the world. Going down from Piazza del Plebiscito towards Via Partenope you reach Castel dell'Ovo, the oldest castle in the city (1st century BC) and also the most spectacular as it is located on the islet of Megaride and connected to the mainland artificially. Here you can enjoy the sunset with views of Vesuvius on one side and Posillipo on the other, you can dine in one of the pizzerias or restaurants in the area or take the elevator to Mount Echia, where you can take fantastic pictures of the panorama.
*The visit of the promenade can be moved to the second day, but I recommend it on both the first and second day as it is one of the most beautiful places in the city.