From Kii-Katsuura (or Shingu) to Koyasan

Hi! This ia Laura and Albert, from Spain.
We are going to Kumano Kodo on the 10th of september.
We want to know which is the fastest way to go from KiiKatsuura/Shingu (or Hongu if it's not possible) to Koyasan!?.
Taxi maybe? Someone knows how to book a taxi?

Thanks so much, we are desperate.

Driving yourself or hiring a taxi would be fastest. The latter would be very expensive (about 30,000 yen).

Your public transportation options are explained on the following page:
https://www.japan-guide.com/e/e4951.html?aFROM=4900_4950

Car hire is good, and the drive is lovely. Depending on which way you go, there are onsen opportunities en route.
I would drive, and look to drop the car in a convenient place in the same prefecture, with a train station, on the way back from Koyasan.
Your other option is as Uji says, or ?maybe? backtrack to Wakayama and then go to Koyasan from there?