You can not go directly to Koyasan Sta. by train. You have to make a transfer at Gokurakubashi Sta. to a cable car.
http://www.nankai.co.jp/global/routemap/nankai.html
There will be only 4 Ltd. Express trains on Koya Line of Nankai to match your hope, if you leave Kurashiki on the same day.
As to a group of 4 members or more, reservation for Ltd. Express trains of Nankai may be made 3 months before the date of ride through a travel agent. Otherwise, wait until one month before the date.
JR reservation can be made from 10:00 of the day one month before the date of ride by Japan Standard Time.

I think you can change from JR to Nankai, not via Osaka Municipal Subway, though it needs transfer at Osaka Sta., which is neighboring to Shin-Osaka Sta. In this way, you will not have to purchase non-JR tickets at least until you get to a station of Nankai.

Via subway:
Shin-Osaka 新大阪 Sta. of JR-West
| Walk
Shin-Osaka 新大阪 Sta. of Osaka Municipal Subway
| Midosuji Line; about 15 min; Fare: 270 yen per adult
Namba なんばSta. of Osaka Municipal Subway
| Walk
Namba なんば Sta. of Nankai

Not via subway:
Shin-Osaka 新大阪 Sta. of JR-West: Track 15 / 16
| Within 3 min.
Osaka 大阪 Sta.: around 10 min. for transfer and wait
| About 10 - 15 min.
Shin-Imamiya 新今宮 Sta. of JR-West: West Gate
| Walk
Shin-Imamiya 新今宮 Sta. of Nankai
Details
- Shin-Osaka 新大阪 Sta. of JR-West: solid map
http://www.jr-odekake.net/eki/img/premises/0610155_3d.p...
Shinkansen trains from the west may arrive at Tracks 20 - 26 on the 4F (sliding to right on this map).
Local / Rapid / Special Rapid trains for Osaka will leave from Tracks 15 and 16 on the 1F (lower right, sliding down on this map).
- Osaka 大阪 Sta. of JR-West: time tables for Nishi-Kujo 西九条
http://www.jr-odekake.net/eki/time_pdf/090314/osaka1.pd...
Trains except those written in red or white digits will stop at Shin-Imamiya Sta.
- Shin-Imamiya 新今宮 Sta. of JR-West
Passing through the West Gate, you will soon find Nankai.