My friend is skeptical about saints and idol worship but is curious about Shirdi. Is there a story about someone like that visiting Sai Baba?

Yes, there is a relevant story in the Sri Sai Satcharitra about a man who was initially averse to idol worship and was a devotee of the formless. As described in Chapter 35, he agreed to visit Shirdi with his friend, Kaka Mahajani, purely out of what is termed a "clinical curiosity." This individual set two very specific conditions before embarking on the journey: he would absolutely not bow his head to the saint, and he would not provide any Dakshina. He made his friend agree to these terms before they set out from Mumbai for Shirdi.