Hi there,
For a 14 day rail pass to be economical, you have to aim to spend more than ¥45,100 on individual train tickets.
Lets take a look at the costs of the pieces of your itinerary:
- Tokyo > Takayama: ¥14300, (275mins, 532km)
- Takayama > Kyoto: ¥9680, (191mins, 314km)
- Kyoto > Osaka: ¥1680, (27mins, 42km)
- Osaka > Hiroshima: ¥9950, (103mins, 345km)
- Hiroshima > Odawara: ¥17530, (286mins, 810km)
- Odawara > Narita airport terminal 1: ¥4600, (107mins, 163km)
TOTAL: ¥57740 for 6 legs
As the total cost of separate tickets is more than the cost of a 14 day rail pass, I certainly recommend you use a rail pass for your trip. You should have a basic saving of around ¥12,640, and will also have the flexibility to change your travel plans should you want, or make extra day trips.
For Hakone I have used Odawara as it is the closest JR station. From here you will need to use the Odakyu rail system to get up to the lakes. I recommend using the Hakone Free Pass for this. For Koyasan, you should use the Nanlkai Dentetsu trains from Namba Osaka. We have a post with more detail on the Koyasan leg here.
Hope this helps!