I agree about the verb, but the main question was: inside or on the inside? I think they're both OK but in this case on the inside expresses better the concept of on the inside surface.
I would stick to the original: "The best colour for a sun umbrella will be black on the top and white on the inside." or, more guarded/reserved: "The best colour for a sun umbrella would be black on the top and white on the inside."