There is no airport on Paxi. You arrive by ferry — usually from Corfu, occasionally from Igoumenitsa on the mainland or Parga on the Epirus coast.
From Corfu: the fast hydrofoil "Flying Dolphin" makes the crossing in about 50 minutes, two or three times a day in season. The slower conventional ferry takes around three hours and runs less often. Both arrive at Gaios.
From Parga: day-trip boats make the crossing in 45 minutes through the summer, usually leaving Parga around 10 am and returning at 5 pm — useful if you only have a day, less useful if you plan to stay.
From Igoumenitsa: a single car ferry leaves a few times a week. The crossing is roughly 90 minutes. The most practical option if you are bringing a car.