There is one Monastery on the island, Zoodochos Pigis. The name means 'Life giving spring'. Google says that its a medieval monastery (1773) and has a few rare icons. There's only one monk there usually with some of his family and a few hangers-on.
I know that from the entrance, up quite a few steps, there is a great view of the bay below and the mountains on the mainland opposite. Women cannot enter wearing men's clothing, wrap around skirts provided in the entranceway. If you enter wearing trews, or shorts for men, and the monk is around then you'll get told off.
There's a spring down below whose waters are supposed to have cured Archbishop Jacovos 11 of lithiasis, gall stones or kidney stones. He decided to build the Monastery where this miracle took place. And Hey Presto, here it still is today. One of Poros's few sites/sights to see