The iPhone maker said the facility, the first of its kind in Europe, would train aspiring mobile app software developers in a programme it hopes to extend to other countries.
Italian Prime Minister Matteo Renzi hailed the planned centre as "an important experiment", and said it would provide opportunities for more than 600 students.
Cook said on Twitter that the company he has headed since its late founder Steve Jobs stepped down in 2011 had created 1.4 million jobs in Europe, a tantalising prospect for Italy, where almost 40 percent of young people are unemployed.
The situation is particularly bad in the poor regions of the south, including the area around Naples.
According to national statistics office ISTAT, in the third quarter of 2015 just 43 percent of working age people in the southern regions had a job, compared with 61 percent in the more economically developed north.
© Thomson Reuters 2016