Cheap Pass to buy during my stay in japan

I will be traveling to Japan from Feb15-Feb24, 2017.
Arrival @Osaka-Kansai Airport. Departure also from Osaka-Kansai Airport.

- Unfortunately I have booked my hotels for 5 nights in Osaka. Plan to travel around Osaka (2days), Kyoto(1day), Nara (1day), Koyasan or Himeji(1day). What pass is best and cheapest to use (Note: I will have to go on a day trip as I always have to return to my hotel in Osaka)? Kansai Thru Pass (3-day Pass) combined with Osaka Amazing Pass (2-day Pass)?

- I plan to go to Tokyo for 4 days (but then, I have to return to Osaka again as I depart from Osaka-Kansai Airport). What is the cheapest and fastest way to get to Tokyo (and vice versa)?

- During my stay in Tokyo, what pass should I use? Tokyo Subway Pass (3-day Pass)? I also plan to go to Mt. Fuji, what's the cheapest way to get there?

- Do I really need to purchase JR Pass before I go to Japan?

Your help is very highly appreciated.

Fandy Susanto

The most effective way you can save money according to what you plan to do it get a 7 day JR Rail Pass (and yes, for your dates you still must buy the exchange order before arriving in Japan) and then bundle as much long distance travel as possible - the trip to Tokyo and back, the trip to Himeji from Osaka, and going to see Fuji (you need to pick Hakone or the Fuji 5 Lakes). The JR Pass will get you partway there, and then an Odakyu or other pass can help you save.
http://www.odakyu.jp/english/deels/
For Koyasan, there is the Koyasan World Heritage Ticket
http://www.japan-guide.com/e/e2358_012.html
In Tokyo the 3 day subway pass is a good money saver.
http://www.tokyometro.jp/en/ticket/value/travel/index.html#anc03
The rest is up to you. The Osaka Amazing Pass might pay off well since it has a number of discounts to some places, but it's unclear which if any you will go to. The Icoca & Haruka is a discount ticket will likely serve you well though.
http://www.japan-guide.com/e/e2359_icoca_haruka.html