The island was known in antiquity.

At the Festival previous to this it is said that Cleomedes of Astypalaea killed Iccus of Epidaurus during a boxing-match. On being convicted by the umpires of foul play and being deprived of the prize he became mad through grief and returned to Astypalaea. Attacking a school there of about sixty children he pulled down the pillar which held up the roof. This fell upon the children, and Cleomedes, pelted with stones by the citizens, took refuge in the sanctuary of Athena. He entered a chest standing in the sanctuary and drew down the lid. The Astypalaeans toiled in vain in their attempts to open the chest. At last, however, they broke open the boards of the chest, but found no Cleomedes, either alive or dead. So they sent envoys to Delphi to ask what had happened to Cleomedes. The response given by the Pythian priestess was, they say, as follows:—
Last of heroes is Cleomedes of Astypalaea; Honour him with sacrifices as being no longer mortal. So from this time have the Astypalaeans paid honours to Cleomedes as to a hero. (Pausanias 6.9.6)

At least one scholar thinks this was an example of chronic traumatic encephalopathy (CTE). Rodolfo Savica, Mario Roberto Vaz Carneiro Filho, Christopher J. Boes,"Cleomedes of Astypalaea: A possible early sufferer of chronic traumatic encephalopathy", Journal of Clinical Neuroscience, Volume 42, 2017, Pages 193-195, (https://www.sciencedirect.com/science/article/pii/S0967586817301881)

In the third century, CE, the Greek collector of sources wrote this:

But Hegesander the Delphian, in his Commentaries, says that in the reign of Antigonus Gonatas, there were such a number of hares in the island of Astypalæa, that the natives consulted the oracle on the subject. And the Pythia answered them that they ought to breed dogs, and hunt them; and so in one year there were caught more than six thousand. And all this immense number arose from a man of the island of Anaphe having put one pair of hares in the island. As also, on a previous occasion, when a certain Astypalæan had let loose a pair of partridges in the island of Anaphe, there came to be such a number of partridges in Anaphe, that the inhabitants ran a risk of being driven out of the island by them. But originally Astypalea had no hares at all, but only partridges. (Deipnosophists 9.63)

At Vathi, in the north east of the island, there is an pre-classical site. "An acropolis was established at the tip of the peninsula, on Cape Elleniko, between the late 4th and early 3rd millennium BC, whose circuit and retaining walls built of boulders are visible today. A square tower with surrounding ancillary buildings was constructed on the peninsula's upper level in the late 4th century BC. Inscriptions however, from the early 6th to late 5th century BC testify to continuous human activity in the region." Andreas Vlachopoulos, quoted in "Vathi on Astypalaia: A palimpsest of the Aegean islands through time"

There is no temple ruins, at least as far as I know. There must have been at least one temple on the island - is it underneith the Kastro or the Chora?

There is, of course, the cemetery for infants. A really remarkable find.