The sink hole cover was copper, but after years of cleaning, the color is wearing off. I'd like it to be the same color in acrylic paint (which I have) as our faucet, which is bronze. Can this be done?
In a word, Yes.
You can paint the metal part to match the faucet; but the finish will be cosmetic, and will wear off with cleaning using anything abrasive. It will not last as long as the original metal-plated surface. You will have to re-apply the acrylic cover-up periodically.
You can get a replacement and/or avoid aggressive cleaning of the used surface-treated one.
Painting will be a short-term hack.