How did Civilization IV build a city on an island?

Yes, you can sail, and then make a paddle boat. You can put immigrants on it and then go to the island to mine.

However, the small island has a large sea area, so it is impossible for the sea to build plots to increase production (except Portugal and the Netherlands, of course, whose UB depends on water). I don't recommend you to open an island city unless there are resources on the island. Early paddle sailing takes time, and you must understand this.