Do I need a JR Pass?

Hi! I will be going to Tokyo from 2 -13 December. I am working around my itinerary with the following places of visit. First stop straight after landing would be to Mt Fuji for 3D2N. I will be staying in the Theme park hotel. I am thinking of going to the Safari and Snow Sown Yeti (all in 1 day) and I would not be doing that while at Mt Fuji. I was told from Mt Fuji to these 2 places which is near Hakone would be faster from Shinjuki. So we would return to Tokyo and then go on a 1 day visit to Hakone. If not, we may bring it to 2 days.

We would also go Yokohama and Disneyland/Sea.

The rest of the stay would be to Asakusa, Shibuya etc. We will be staying at Shinjuku.

Given this rough itinerary.... would it be better to get a JR Rail pass or we have other cheaper, if not faster travelling options?

Thank you!