to be honest a day for Capri Island is not enough. Capri is a lovely island with you will need to spent more than a day may be spending 2 days in Capri is better option.
We been to Capri Island beginning of Ocrober, the day we went to Capri weather was nice but a day before it was raining. Few must do things in Capri:

Take a private boat tour to see around the Capri island, there are few grottos that you must see most popular one is Blue Grotto colour of the sea is amazing aqua blue. Even it is october you must wear your swimming costume and you must swim in Capri Island.

I tasted one of the best icecream and pasitery in Capri it is called Buonocore Gelateria you must visit this place, dont forget to buy their pasitery with lemon and almonds biscotties they are amazing.

Villa Verde is another place that you must visit, perfect Restaurant, they have an interesting history as most of the famous people are eaten there, restaurant has delicious menu and food.

capri