Cu(i) and Cu(II) differ in the electron configuration of the outer shell. Normally, all 3d-states (10) in pure Cu are occupied. In Cu(i) the 3s electron is taken to fill the outer shell of oxygen and in Cu(II), a 3d electron fills the oxygen shell. This can be tested by absorption measurements. If you have nearby a synchrotron radiation center, you can test the following by XAS (X-ray absorption spectroscopy):
In Cu(II) you can excite a 2p-electron into the 3d-state (a strong absorption line nearly at 930 eV). In Cu(I) this absorption is weak because the d-state is occupied. I myself tested this with my copper sample holder: A fresh polished copper block exhibited no absorption but an older oxydized Cu-block absorbed. (The best absorption of course exhibited Cu-O2 planes in high Tc superconductors).
Of course the best method is XPS, however if you want to have a general idea, you can impregnate your solution into cellulose (simple filter paper) and track the color change: Cu+ is almost colorless, Cu2+ is blue.
(I believe, cellulose will provide somewhat similar conditions as activated carbon)
For XRD you need to have a crystalline structure. Depending on your precursor and drying conditions most likely you will have mixtures of crystalline structures and it will be very difficult to say what you have on the surface. Besides, the concentration will be small because of carbon and you will not get good XRD patterns. If you worry about Cu (I) oxidation, you can keep your impregnation solution acidic, or even you can add ascorbic acid which provides a reducing conditions for Cu, and prevents its oxidation.