The daily life of the Italian noble in your example would be governed by the three vows he took upon entering the Order: chastity, poverty, and obedience. He held no property, everything belonged to the Order as a whole. He could not be married or have sexual relationships. He had to abide by the Order's rules, and could be physically punished if he disobeyed.

As a noble fighting for the Order, this man would be a knight of the Order, rather than a sergeant-at-arms (non-noble soldiers), a sergeant-at-office (administrative brethren), or a chaplain. To become a knight he had to prove that both his parents were of gentle birth. For seven years after doing so, any other Hospitaller could challenge his legitimacy.

As an Italian, he would live on Rhodes in or near the auberge of the Italian langue. The langue was one of seven (eight after 1462) national divisions of the Order and the auberge was the langue's headquarters on Rhodes (and later on Malta). The other langues were the langue of England (which also included the Order's Scottish, Irish, and Welsh brethren), Provence, France, Castille-Portugal, Aragon (later Aragon-Catalonia-Navarre after 1462), Germany (which included Scandinavia and Poland), and Auvergne.

The daily life of a Hospitaller on Rhodes or Malta was spent primarily in the company of their langue. He would eat and pray alongside the brethren of his langue. If the island was attacked, he would defend the area of the walls assigned to his langue. Each day the brother had to perform the seven canonical hours (rather than the normal nine monastic hours). He would wake shortly before midnight when he would join his brethren in the langue's chapel to sing the first hour, lauds. He would then go back to bed only to wake again before dawn to sing the next hour, prime. Further church services would follow at 9:00, noon, 15:00, 18:00, and 21:00. Brethren also had to recite 150 paternosters a day. The rest of his time would be spent patrolling, manning the defences, training, attending meetings of the langue, or managing any land or property he had been assigned by the Order.

Each Hospitaller was paid by the Order to provide for their food, shelter, clothes, etc. These payments were called soldea, which was 50 or 60 Rhodes florins a year, or goods of equivalent value, and tabula, which was 20 florins a year. The knight would pay some of this money to the langue's auberge, to pay for his meals there. When a langue was short of money he could also be pressured to pay to help fund repairs to the auberge or to the langue's section of the island's defences.

The sixteenth century saw some big changes for the Order. Firstly, the brethren left Rhodes in 1522, surrendering the island to the Turks. After several years in Italy, the Hospitallers settled on Malta in 1530 and began a new capital there, Valletta, in 1566. The auberges were rebuilt, first at the Order's initial Maltese headquarters of Biru, and then in Valletta. Whilst this Italian knight-brother would still eat and worship communnally, from 1571 until the Order's expulsion from Malta in 1798, he would have had much greater contact with non-Hospitallers than he ever had on Rhodes.

On Rhodes the brethren had resided in the collachium, a district around the island's castle where only Hospitallers could live. This had kept the brethren and the locals somewhat apart, but when the Order moved into Valletta in 1571 no collachium was established. Contact between the Hospitallers and the locals increased. This became something of a problem for the Knights. In 1668 an inquisitor complained that because the Hospitallers no longer lived communally and instead could live anywhere on the island, they had become glutonous, a violation of their vow of poverty. This erosion of the monastic aspects of the Order continued into the eighteenth century when one knight, Camillo Spreti, warned newly arriving knights to beware of falling for the local women. After Valletta's theatre opened in 1732, the Hospitallers themselves began to attend, whilst a visitor witnessing a Hospitaller galley leaving on campaign in 1770 claimed that the thirty knights aboard were waving at their mistresses as they left.

In terms of military service, your Italian noble had two options open to him, both of them called a caravan, a tour of duty which lasted six months. Either he could serve on a ship or he could help garrison one of the Order's other islands or forts. If the knight was stationed on land, then his daily life would be little different than back in Rhodes or Valletta. He would continue to mix mostly with the others of his langue, his religious schedule would continue, he still had to sing the hours and recite 150 paternosters a day, eat with his brethren, and patrol a series of defences. This type of service could be performed at anytime of the year.

In contrast, a naval caravan was carried out during the sailing season (spring to autumn). The knight's daily life, again, would change little. He would still eat communally and be expected to celebrate the canonical hours. Every ship had a chaplain to maintain religious discipline. The main difference would be a greater chance of combat than when garrisoning a castle. Naval caravans would attack the ships of the Ottomans, Barbary Corsairs, and any other Muslim power they encountered. They could also carry out island or coastal raids.

After serving three years at Rhodes or Malta and performing three caravans (four in the eighteenth century), a Hospitaller would be eligible for promotion. This promotion could be an administrative position at Rhodes or Malta, or managing one of the Order's farming estates in Europe. If your knight had performed several caravans, served many years in the Order (as promotion was also based on seniority), then by the end of his life he probably would have been in charge of one of these estates, called preceptories. Life on the estate would be much quieter than at Rhodes or Malta. Most preceptories only had two or three Hospitallers. Some only had a single brother to manage them. There would also be secular servants who farmed the land, collected taxes, etc., but in his final years your Italian nobleman would rarely eat or pray surrounded by his brethren. His life at this stage would be focused on farming and developing the estate in order to send money to Rhodes or Malta. He would also attend general chapters of the priory, which was the headquarters that each region's preceptories reported to. Unless a local conflict broke out or if the knight involved himself in politics, it was unlikely that he would see combat again as a preceptor, particularly after the fifteenth century.

p.s. Though I mentioned that the Hospitallers had vowed celibacy, this wasn't always adhered to, so your example Italian noble may have had a lover or visited prostitutes in the town of Rhodes or in Valletta on Malta. This became such an issue that in 1581 the grandmaster expelled all prostitutes from Valletta to the outlying villages. In July that year the rest of the Order deposed him.

Sources:

Emanuel Buttigieg, Nobility, Faith and Masculinity: The Hospitaller Knights of Malta, c.1580-c.1700 (London, 2011).

George Cassar, 'Malta and the Order of St John: Life on an Island Home', in Emanuel Buttigieg and Simon Phillips (eds), Islands and Military Orders, c.1291-c.1798 (Farnham, 2013), pp. 75-83.

Gregory O'Malley, Knights Hospitaller of the English Langue 1460-1565 (Oxford, 2005).

Patricia Micallef, 'The Vision of the Island of Malta and its Role in the Transformation of the Order's Mission as Seen by the Seventeenth- and Eighteenth-Century Traveller', in Emanuel Buttigieg and Simon Phillips (eds), Islands and Military Orders, c.1291-c.1798 (Farnham, 2013), pp.115-25.

Helen Nicholson, The Knights Hospitaller (Woodbridge, 2001).