After some thought, I realized that Naoi having any cash left on her or not is irrelevant due to a certain detail in the chapter. Her dad asked her to "withdraw" some more if she didn't have any on her at the time, so to do that so late at night, she'd need her debit card. A love hotel would accept debit.
As for why her dad was so insistent on cash, in Japan there are still numerous Izakaya's that are Cash Only.
As for love hotels, there is little to no interaction with the staff. They don't show people to their rooms. Either it's a machine like slyss pointed out, or there's a window with a privacy screen where the guests pay without the employee ever seeing what they look like. The only other people one might see in a love hotel are other couples in the common area hallway leading to the rooms.