Accommodation on Koyasan comes in form of temple accommodation rather than ryokan:
http://www.japan-guide.com/e/e4905.htmlIt is a very popular thing to do for those who do not mind spending a night in simple conditions with shared toilet, sink and bath and cold temperatures in winter.
Then again, if this is your first visit to Kyoto, I recommend to spend at least two full days in the city in order to appreciate it enough. Depending on the details of your itinerary, that could make it difficult to fit in Koyasan.