According to the places you wish, you want to be the centre of Kyoto to leave to Osaka.
Your route is kinda complicated, so I recommend an other option to save your time.
Start from Ookitayama-haradani-inuicho -> Arashiyama Banboo Groves ( approx 30 mins by car, 1 hour by bus).
-> Fushimi Inari Shrine ( approx 1 hour by trains)
-> Nishiki Market ( approx 30 mins by train)
->Kinkakuji (approx 30 mins by bus)
-> through Gion street and Higashiyama area ( take a walk, there's a walkway from Kinkakuji to Higashiyama district going through Kiyomizu Temple and Yasaka Shrine, then you'll be Gion district. The walkway is called "Tetsugaku-no-michi")
Arashiyama and Fushimi Inari is very far from the heart of Kyoto, and each other as well.
So maybe taxi would be the better way to save more of your time.
And your concern of the transportation.
There's Bus-pass that allows you to take all city buses for 1 day ( within the limited area).
Also, if you have chance to take buses and trains as well, there's a One day pass as well.
You can get it at Kyoto station, subway station, and maybe at your hotel.
This website will help you for the transportation.
If you will take some public transportation in Osaka too, I recommend you to get Pitapa.
It's a chargeable IC card that you can use for buses, subways and the other trains, with touch-and-go only.
So you don't have to keep the coins.
1. If you take more than 3 times of buses at least on the same day, get Pass.
2. If you don't take much public transportation, get Pitapa.
Always welcome for more questions!
Hope your trip comes much more enjoyable!