I suggest...
(option 1: using JR
Day 1: Singapore to Osaka
Day 2: Osaka (Universal Studios)
Day 3: Osaka (explore rest of Osaka)
Day 4: Day trip to Kobe, stay at Osaka
Day 5 (
JR Kansai Area Wide Pass day 1, 7000yen for 4 days) : Day trip to Nara, stay at Osaka
Day 6 (JR Kansai Area Wide Pass day 2) : Day trip to Okayama, stay at Osaka (Kurashiki, Okayama castle, Korakuen)
Day 7 (JR Kansai Area Wide Pass day 3) : Osaka to Kyoto, stay at Kyoto (Arashiyama, Fushimi Inari)
Day 8 (JR Kansai Area Wide Pass day 4) : Day trip to Hikone, stay at Kyoto
Day 9: Kyoto
Day 10 (
JR Kansai Area Pass 1 day 2000yen for Haruka Express non-reserved seat) : Kyoto to Kansai Airport (via Haruka Express), then Kansai Airport to Singapore.
Or if you don't mind spending more on transport (getting
7day Nationwide JR Pass, 28300yen):
Day 1: Singapore to Osaka
Day 2: Osaka
Day 3 (JR Pass day 1) : Osaka to Hiroshima, stay at Hiroshima
Day 4 (JR Pass day 2) : Day trip to Miyajima, stay at Hiroshima
Day 5 (JR Pass day 3) : Hiroshima to Okayama
Day 6 (JR Pass day 4) : Okayama to Kobe
Day 7 (JR Pass day 5) : Kobe to Takayama (via Nagoya) *either go Shirakawa-go from Takayama on this afternoon or following morning. If you take 6:51am Shinkansen out from Shin-Kobe station, you should be able to catch Wide View Hida Ltd Exp train #3 to Takayama from Nagoya and arrive at Takayama at 1055am
Day 8 (JR Pass day 6) : Takayama to Kyoto (via Nagoya)
Day 9 (JR Pass day 7) : Kyoto
Day 10: Kyoto to Kansai Airport
(If you want 1 more day at Kyoto, then skip the extra night at Hiroshima, or maybe give up either Okayama or Kobe)
Or you can maybe see
my Japan (Jul 2011) thread for more ideas since I went to Hiroshima/Miyajima/Kobe/Osaka/Takayama/Kyoto on that trip.