Hi Stella Ong,
I would recommend using a 7 day JR Pass for your travels. Travel from Tokyo - Kyoto - Osaka - Hakone/Tokyo is the most expensive. So ideally you cover this using the JR Pass.
For 12 days, you could do the following:
Day 1.) Narita Airport - Kyoto, travel on the same day and activate the JR Pass. This way you have most travel behind you and you can start focusing on enjoying Japan. You will also cover travel from Narita Airport using the JR Pass.
Day 2-3) Explore Kyoto
Day 4.) Day trip to Nara.
Day 5.) Travel to Osaka (only 30min away from Kyoto) and explore Osaka.
Day 6.) Osaka
Day 7.) Osaka and travel to Hakone (Shinkansen train to Odawara station).
Day 8 and 9.) Use Hakone Free Pass to explore Hakone. Can also include a visit to closer to Fuji, such as Fuji Five lakes.
End of day 9, travel to Tokyo.
Day 10-12.) Explore Tokyo and return to Narita Airport.
Hope this helps,
Daniel