Yes, you can use the JR Rail pass to get to the Hakone area. The JR Rail pass can be used on JR Trains and the Shinkansen (Bullet Train) only. You can get to Odawara Station using your JR Rail pass, but from there you will need to pay for transportation separately. Getting to Odawara Station from Tokyo only takes about 30 minutes and about 2 hours from Kyoto. Hakone is a short 15 minute bus or train ride from Odawara Station.
If you have a JR Rail pass, take the Shinkansen to Odawara. At Odawara you can by a Hakone Freepass that will cover your travels in the area. The Odakyu Line also runs directly from Shinjuku to Hakone-Yumoto, but is not covered by the JR Rail pass. You can purchase the Hakone Freepass to include travel to and from Shinjuku Station. The Hakone Freepass makes your time in Hakone efficient and economical. You can ride the different methods of transportation just by shoeing your pass, not fidgeting with change and money.