At Mount Koya you can't miss the chance to have a Koyasan-style vegetarian lunch.
I recommend the restaurant Chuoshokudo Sanbo which is located in a side street of the main street (look for it with googlemaps and you will have no trouble finding it). A selection of saucers based on vegetables and tofu typical of Koyasan will be served.
For dinner, if you fancy a great ramen you have to go to Tsujimoto ramen. The recently opened restaurant is small but very welcoming. The staff (ask for Maru) will know how to pamper you by offering you small samples while waiting for the excellent ramen.