Υou can get to Patmos, by ferry, catamaran or local small boats.
They all connect to the islands which are located nearby, such as Samos, Leros & Kos, where there are airports, too.
Useful Links
BLUE STAR ◊ ANEK (Kalimnos boat) ◊ DODEKANISOS SEA WAYS ◊ HELLENIC SEAWAYS