Tokyo, Hakone, Kyoto, Nara, Osaka — in that order, for reasons of luggage and energy.
Days 1 to 3, Tokyo
Arrive, and do not plan anything demanding for the first afternoon. Then one day east — Asakusa, the Sumida river, Akihabara — and one day west — Shibuya, Harajuku, Shinjuku at night. Get a Suica card at the airport.
Day 4, Hakone
Forward your main luggage to Kyoto by takkyubin and travel with an overnight bag. The Hakone loop — pirate ship, ropeway, sulphur vents — ends at a ryokan with an onsen and a view of Fuji if the weather cooperates.
Days 5 to 7, Kyoto
Fushimi Inari at 7am before the crowds, Arashiyama's bamboo the same way on another morning. Kinkaku-ji, Nijo Castle, and one evening walking Gion and Pontocho. Three nights is the minimum.
Day 8, Nara
A day trip. Todai-ji's great buddha hall is the largest wooden building most people will ever stand in, and the deer are genuinely everywhere.
Days 9 and 10, Osaka
Dotonbori for the food, Osaka Castle for the morning, and Kansai airport twenty-five minutes from the city. Flying home from Osaka rather than backtracking to Tokyo saves most of a day.
On the rail pass
The nationwide JR Pass is no longer automatically the cheapest option for this route. Price individual shinkansen tickets before assuming — we do this for every Japan booking.