there's a one square island just next to my coast and soon it will be inside my cultural borders, and since it has rubber on it, the question is, is there a way I could use that rubber without building a city on that one square island?
Nope. You need a city there. Good thing is that the rubber will be automatically connected to your city (no roads needed). Then you just need a harbor and it's connected to your capital (assuming your home continent has at least one harbor). Also that city is totally protected from enemies (except against marines).