Is JR Pass recommended for my itineary?

1st day : Kansai Airport - Shin-Osaka - Namba (Midosuji line)

2nd & 3rd Day : Tour in Osaka (Umeda, Nipponbashi, Ebisucho, Tenjinnashi-suji rokuchome stn, Minami (Namba), Tennoji

3rd&4th Day : Tour in Kyoto(Arashimaya, Inari, Tofukuji, Kitaoji, Demachi-Yanagi, Keage , Nijojo-mae, Marutamachi, Shijo stations). Bus transport is maybe included also.

5th Day : Taking Shinkansen to Tokyo - Asakusabashi - Tour in Tokyo

6th - 9th Day: Asakusa, Shinjuku, Yoyogi Ikebukuro, Akihabara, Tokyo station, Shibuya, Roppongi and Mt Fuji (Lake Kawaguchiko).

Last Day : We might be taking Keisei Skyliner from Ueno/Nippori to the airport

No, a JR Pass will not pay off.

So it is recommended to use local transport card and buying shinkansen ticket separately?

Btw, if I want to visit Mt Fuji (Five lakes and Hakone) , are Fuji Five Lake Pass/ Hakone worth it?

yes jr pass will not cost as much as buying separate tickets.

whether a fuji hakone pass will pay off is dependent on your itinerary. it only covers one return trip from tokyo. so if you plan to do them are separate day trips without overnight stays. then you are probably better off finding regualar transport and pay for daily tickets.

4000 yen for a hakone free pass from odawara.
3350 yen for a fujikyu train/bus pass from otsuki.