It's not like Kagome doesn't believe in demons, or ghosts, or spirits, exactly. She knows that lots of people don't, these days, and she's never seen one, but she also knows that her family has guarded their ancestral shrine for hundreds of years, and they probably had some reason to do that. Maybe it's motivated reasoning, but she doesn't want to believe that her family's ancestral obligations are completely pointless. So probably there some are spirits out there, somewhere.
Still, Kagome figures that they all mostly keep to themselves, and that there aren't really very many of them. Otherwise you'd see them about, sometimes, and you never do.
When she goes to help Sota find the cat that's wandered into the shed where the well is - she doesn't think the noises coming from the well are really a ghost. She's not five; she knows that even if there are ghosts about, somewhere, you don't just run into them in dark corners of your family's shed, especially if you already know that there's a cat messing around in there. If that sort of thing happened with any regularity, then more people would believe in it. Especially if the ghosts were generally inclined to do anything besides make creepy noises, which they're probably not. So there's nothing to be scared of, really, and she doesn't want to look dumb in front of her nephew.
She does, in fact, find the cat. The cat is not in the well.
She spends about three seconds wondering what was making the noises, which is all the time she has before the cover over the well breaks open.