I was really surprised how much I loved Corfu. Depending on if your ship gets in early enough, I would buy a museum pass at the Old Fort (which is right near where the shuttle buses drop you off or you can walk) and visit most of the museums. They're scattered around Corfu Town so you'll get a flavor of the town just by walking. Some of the museums close at 3:00 so it's not worth it to buy a pass if you get in late. Make sure to visit the English church which is at the Old Fort. The Palace houses a large Asian art collection.

Finally, get a drink or an ice cream and sit on the Esplanade which is just an incredible spot to people watch.

If none of this interests you, I would take the recommendation of renting a car and traveling to Pandokrator as previously recommended.