Santorini (Thira) International Airport: Emporio is about 10 km south of the airport. The actually sensible option with bags or a late flight is a pre-booked taxi/private transfer: roughly 15–25 minutes, usually around €30–45 depending on season and time. There is no useful direct airport-to-Emporio bus; the cheap route means going via Fira and changing onto a south-island service, which can easily take 60–90 minutes once waits are included.
Athinios Ferry Port, Santorini: For Emporio, take a pre-booked transfer or taxi: about 20–30 minutes, with fares commonly around €35–50 in high season. The public option is the KTEL bus from Athinios to Fira, then a second bus toward Emporio/Perissa; it is cheap but ferry arrivals create long queues and the connection is not elegant with luggage. Do not assume a taxi will be waiting—Santorini has too few of them.