Patmos has no airport however it has a heliport for those who want to get there in one and a half hours from Athens.
There are also Blue Star ferries which depart from Athens (Pireaus) which take 7 to 8 hours, however the ships are modern and well organized. The restaurant in first class is very good. We strongly recommend you book a cabin.
Another option is to fly to one of the neighboring islands, such as Samos and Kos which have international airports. Kos especially has good connections to Patmos.
Please ensure you are aware of connection times between islands so that your flight coordinates with the boat.