To pass or not to pass

Hello,

My partner and I are travelling to Japan for the first time, and would like some feedback to decide whether we should buy a pass or not.

Our itinerary is below.

We are thinking of the following:

Buying some kind of West Japan pass for the first few days, and then buying single tickets.

OR

Activate a 7 day rail pass on the 15 of October, to cover for our longer distance trip from Koyasan to Tokyo via Hakone.

Thanks a lot for your help!

12 October - Kyoto -
13 October - Kyoto -
14 October - Kyoto -
15 October - Kyoto - day trip to Nara
16 October - Mt Koya -
17 October - Mt Koya -
18 October - Hakone -
19 October - Hakone -
20 October - Tokyo -
21 October - Tokyo - Nikko
22 October - Tokyo
23 October - Tokyo
24 October - Tokyo - Fly home