Santorini is the in the southern Cyclades island group and Patmos is in the northern Dodecanese group of islands, near Turkey. A day trip is not realistically possible. You need to look at a map of the Aegean Sea. Santorini has Akrotiri, ruins of the Bronze Age Minoan civilization. Or go to Napflion on the Peloponnesean Peninsula where you are surrounded by amazing antiquities, a real treasure trove. And, of course, Athens is filled with them too.
The Cycladic islands are the most visited of the Greek island groups. If you were traveling to Rhodes or Kas in the Dodecanese Islands you could visit Patmos as a day trip. Santorini is usually only suggested as a day trip from Crete. You could fly Santorini to Athens to Rhodes and ferry to Patmos.