The guidebooks tell you that for travellers with a Japan Rail Pass to get to Koyasan you should take a long winded route to Hashimoto (Wakayama) which takes about 2 hours from Osaka station
Everybody then has to pay 850 yen to take the Nankai line train to Koyasan - since the line isn't owned by JR.
However, if you use your Japan Rail pass to get to Shinimamiya (about 17 mins from Osaka station) and then take the Nankai line from there - it costs 1260 - just 410 yen more than you would be paying anyway from Hashimoto. And from Shinimamiya it takes about 46 minutes to Hashimoto - so about 70 mins from Osaka station (with a reasonable connection time).
So for the extra 410 yen, you save about 50 mins. And since there is a gap of a couple of hours in the service to Koyasan around lunchtime - that 50 mins may make an ever bigger difference !
Be aware that on the Nankai line - the 1260 fare doesn't allow you to use the "limited express" trains to Hashimoto - which require an additional 780 yen fee ("express" trains are marginally slower and don't require a supplement). The extra fee may not save you any time - it just may mean you don't need to change trains at Hashimoto. But it's a small station, so changing trains isn't a problem.