There is no airport in Amorgos. The closest airport is on the island of Naxos, in the Cyclades, which serves daily domestic flights from Athens.
The flight time from Athens to Naxos is 40 min. From the port of Naxos, there are daily ferries to Amorgos.
Alternatively, you can reach the airport of Santorini which received international flights.
Find the flights to Naxos and the flights to Santorini.