The recommended hotel is very close to Puerta del Sol and Plaza Mayor areas. You can start or end your days visiting them. Aim to arrive in Madrid in the morning so you can have the afternoon of your first day.
For Day 1, if you arrive before check in time, the hotel will allow you to leave your luggage with them. So you can enjoy Mercado de San Miguel (food market) and the Palacio Real de Madrid, the city’s Royal Palace.
12PM – Mercado de San Miguel (food market)
2PM – Royal Palace of Madrid and Sabatini Gardens
6PM – Dinner at Traga Tapas
For Day 2, I recommend Toledo and Segovia. You can book a day tour for this.
For 3, I recommend visiting Retiro Park and the Paseo del Prado, home to Madrid’s 3 most famous museums. You can check out early and leave your things at reception and go to the museum. Before heading back, San Gines and their famous chocolate and churros.
Remember to give yourself ample time for the travel back to the airport, checking and passport control!