Booking Eko-in Temple Koyasan

I booked to stay at Eko-in for a single night through their website, I received an email confirming my reservation however they didn't ask for a deposit or credit card details. I'm just worried because on sites like Japanican or Booking.com they often ask for credit card details should I not show up and they can charge me, the official Eko-in website didn't ask for any of those and yet claim that they will charge me should I cancel to soon to the arrival date etc... The real question is that when I arrive at Koyasan, will there be a room for me to stay in? Has anyone else booked this way?
Thanks!

I have booked online at a different place, and my card was charged immediately. Obviously they can't charge you without your card number, so only they know what is going on in their heads. But the same things goes for booking at a business hotel like Toyoko Inn - for the English site they want your card number when making the reservation, but not on the Japanese version.

One other thing you might consider when picking a place, and this matters if you want to see the cemetery at night and particularly in the summer months which have longer daylight hours. Some temples have curfews (often 9 PM), so you might not be able to go, or even just go outside and look at a billion stars at night if the weather cooperates.
There is one place that does night tours
https://www.night.koyasan-okunoin.com/
but of course they charge for it and there is nothing stopping you from just going yourself. They do however list the temples with the curfews at
https://awesome-tours.jp/en/tour/night-tour-return-bus/
Yours appears to be unlisted so you're fine, but if it were they say they have a free shuttle - nevertheless it would still be constricting to have to leave so early.

Not uncommon in Japan. You pay (usually in cash) when you get there, either at check in or check out.

I have a reservation there and I haven't given any card details either. I dropped them an email (found on their website) and they confirmed my reservation so it seems it's just a pay at check in kind of place.