Hi Monique-chan,
The shuttle bus which is hotel offers looks like a good deal to me, it's about the same price as a normal train ticket and you can combine it well with a 7 day JR Pass.
Getting a JR Pass would be a great addition to your trip, you can use it to travel to Kyoto, Nara, Nikko and even Hiroshima. A 7 day JR Pass is just about the same price as normal tickets for a return Tokyo - Kyoto, this means that any additional travel is what you save. This can really add up if you use it to make side trips to Nikko for instance.
Here's a little break down of ticket prices:
Hakone (Odawara station) - Kyoto ¥ 11,750
Kyoto - Nara return ¥ 1,380
Kyoto - Tokyo (Shibuya) ¥ 13,520
Tokyo (Shibuya) - Nikko (return) ¥ 9,640
This adds up to ¥ 36,290 the 7 day JR Pass is ¥ 28,300 so you would save at least ¥ 7,990 you can also use to JR Pass for a potential visit to Hiroshima and Miyajima (which alone would save you an other ¥ 22,720). In addition you can also use the JR Pass on the JR Network in Tokyo. I would recommend using the JR Pass from day 4-10, this way you can really make the most of your pass.
On day 3 when you travel to Hakone, take a direct train from Shibuya to Shinjuku, here you can board the Odakyu line to Hakone and travel on to Gora, consider buying a Hakone Free Pass for this journey as it covers all kinds of transport in Hakone and gives many nice discounts.
On day 4, from Gora take the Hakone Tozan Railway to the station Hakone-Yumoto, from there take a local train to Odawara station from where you can catch a bullet train directly to Kyoto.
On day 6, I would certainly recommend making a day trip to make the most of your time in Japan (and your JR Pass). Hiroshima is a great visit, the peace memorial museum is not to be missed. In addition you can make a side trip to Miyajima which is one of the most iconic places in Japan.
On day 9, when you visit to Nikko, take a train to Tokyo station here you can board the Shinkansen (bullet rail) to Utsunomiya, here you can transfer to the Nikko line bound for Nikko.
- In Tokyo consider a visit to Odaiba which is a very modern part of Japan had is a very interesting area for shopping and other attractions. Tokyo is very large and offers countless areas of interest, let me know if you have any special interest and I can give you more specific recommendations.
Hope this helps!
Daniel