There is one water fountain airside in Naples airport: take a sharp left as you exit World Duty Free and you will see a water fountain sign high up on the wall (the actual water fountain was obscured behind shelving when I visited on 8 April 2018). I did not look for water fountains before going airside.
Update August 2019:
Above described location is still correct. When exciting the duty free shop keep a bit left towards the restrooms. There’s a sign at the wall (not the ceiling) showing the water fountain. However, it’s only a small amount of water coming out and it’s not equipped to fill bottles with water.
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