The Italy international, who joined Chelsea from Atalanta over the summer following an award-winning loan spell at Cagliari, addressed reporters from Sky Sport at the glamorous event held at the Teatro alla Scala.
"I'm doing well, it's still a matter of a few more days but I feel a lot better," Palestra said when asked about his fitness.
Asked how he was settling into life in west London, the 21-year-old was full of praise for his manager and new teammates.
"From the very first day, the boss Xabi Alonso and my teammates have made me feel great. A lot of things have changed but I like everything here," Palestra said.
"Am I a starter? In a squad like this there are no undroppable players."
Palestra arrived at Stamford Bridge for a fee in the region of £47m after being named Serie A's Defender of the Year following his loan spell at Cagliari, where he made 37 league appearances last season. His versatility across both full-back positions has been highlighted as a key asset within Alonso's system, giving the Spanish head coach flexibility along the flanks as he continues to shape his squad following his appointment.
Palestra was also asked whether there was a determination within the Italy squad to return to major tournaments, after the Azzurri missed out on a third consecutive World Cup earlier this year.
"What I can assure you is that we all had our sights set on big goals. We have a wonderful group, and I'm sure that by working together we can achieve great satisfaction," he said.
The defender's rapid rise has been one of Italian football's standout stories of the past year, progressing from a loan spell in Serie A to a big-money move to the Premier League and full international recognition within the space of a few months.
With his recovery from injury now nearing completion, Palestra will be hopeful of adding regular first-team football for Chelsea to that list of achievements before too long, as he looks to establish himself as a fixture in Alonso's plans for the season ahead.