You can use contact cement to adhere copper sheets to wood cabinets, but make sure surfaces are clean and completely dry/free of dust.
There are videos demonstrating the ashesion of copper to wood. I would ot advise making the copper as wide as the cabinet, due to transfer of body oils to the metal. An insert would be a better alternative.
There are copper looking back splash sheets you could use or even ceiling tiles of the same. They come in stick and peel and are real nice looking. That is what I would use.
I'd use a 'synthetic' copper TONE product ... you won't have a 'tarnish/polish' issue