Airplane schedules:
There is no air connection to Patmos island.
You may fly to Samos, Kos or Leros and from there catch a boat to Patmos.

Ferry Boat and hydrofoil information for Patmos:

There are 10 weekly scheduled departures from Piraeus to Patmos in the summer and 4 in the winter.
Travel time from Piraeus is approximately 10 hours.