Cu+ and Zn2+ are isoelectronic. Cu+ will not be magnetically active or Jahn-Teller active (as Cu2+ is), so the coordination geometry should be roughly the same as for the zinc coordination, and nitrogen NMR should not be significantly affected (between Cu and Zn). The only thing I would expect to see change is the equilibrium bond length between the histidine N and the metal cation, with a shorter bond length for the divalent cation, based solely on electrostatics.
Maybe I'm oversimplifying, but it doesn't seem that there should be a huge difference. Just remember that Cu+ is very easily oxidized.