The Ordinances requiring the antigen test on arrival in Sicily for the Netherlands, and a few other countries, are valid up to September 1, 2021 inclusive. There may or may not be a new ordinance in the next day or two, extending the tests for a further period of time. This ordinance would be available in PDF format on the Sicily Region website below. So, you could check there as time goes on. Note that the Palermo Airport website may not be updated as quickly as the Sicily Region website. There is no requirement to do a test when leaving Sicily, but check the Netherlands government website in case they require a test.
The Italian version of this site has the most recent info:
Sicily Region Covid ordinances:
The Antigen tests for arrivals at Palermo Airport are still in operation until September 30, 2021 inclusive for the same 7 countries, regardless of whatever test results or vaccination status you may already have.Edited: 01 September 2021, 03:48