You are correct that during this time in October many places begin to close as the weather turns cooler.
Prince Edward Island is a different place in October than at the high time in August - however fall is one of my favorite times in our province. The streets are quiet and driving in the country is beautiful as the all the leaves turn colour, not to mention all the incredible fresh food that has just been harvested. You may want to focus your activities around Charlottetown, as more of the businesses there are year round operations - then take day trips around our Island to explore the scenery.
While most summer attractions are closed, the galleries, museums and local artisan stores are still open. In Charlottetown I suggest checking out the Confederation Centre Art Gallery, it is a wonderful national art gallery located in the centre of downtown. The centre also houses an exhibit on the first meeting of the fathers of confederation here in Charlottetown. I also love to explore Beaconsfield House, a house from 1877 that is restored and operated as a museum by the province.
There is always lots to see and do in Charlottetown but a few suggestions for other Island spots in the fall I would visit the Anne of Green Gables Heritage Place in Cavendish. In addition to exploring the era of the early 1900s and the world of Lucy Maud Montgomery they also have a wonderful wooded trail to explore on the site. This is gorgeous in October.
You can always visit Glasgow Glen Farm, where they make a fantastic local gouda cheese. While on the north shore I would head to the Dunes Studio and Gallery. It is a wonderful art gallery and pottery studio. We have visited many time and always enjoy it. They also have a wonderful café there for lunch and dinner.
So you can see that there is still lots to do here in October, but it is best that you check the website or call ahead for any place you find interesting, as each seasonal business has its own closing date.
Hope you enjoy Prince Edward Island in the fall as much as my family does.