Producing colouring of maximal planar graphs G from colouring of dual of G
The simple but useless answer is that I don't know of such a scheme. However, more to the point: proving that such a scheme worked would be tantamount to proving the 4-color theorem. It is very easy to prove that duals of maximal planar graphs have 4-colorings (in fact 3-colorings except for $K_4$ by Brooks' theorem), so if you could prove that a 3- or 4-coloring of the dual graph could always be converted into a 4-coloring of the primal graph then you would have also proved that a 4-coloring of the primal graph exists.

Perhaps there are more, but the only widely-accepted alternative proofs of the 4-color theorem reported by the Wikipedia article are the ones by Robertson, Sanders, Seymour, and Thomas (in connection with a faster algorithmic version of the theory) and Werner and Gonthier (in connection with a formalized proof in Coq). I think both are in roughly the same style of Appel and Haken, of proving that a reducible configuration exists, rather than using dual colorings.