Rough Itinerary in Kansai region

Hi everyone, my name is Allan and i am from Singapore. I have a 9d8n trip booked to kansai region (kyoto-osaka-nara) in Dec and i have some questions about my rough plan.

Day 1-4 : Kyoto
Day 5: Nara day trip
Day 6: Leave Kyoto and go to Osaka. Once reach osaka, go to Mount Koya for 1 night temple lodging
Day 7: Leave Koyasan and return to Osaka
Day 8: Osaka
Day 9: Return to KIX

Here are my questions:

1. Solo traveller, need recommendations on budget hotel in kyoto and osaka. especially for osaka as my return flight is at 11am and i will be going to KIX from osaka. i have to be at KIX by 9am so its quite rush. Would prefer hotel that is convenient to return to KIX.

2. I have a problem with my day 6 and 7. As you can see for my day 6, once i reach osaka i will not check in first to my osaka hotel. i will go to Mount koya as i will be overnight there. What can i do for my luggage as i am not sure if i can put my luggage at osaka hotel until then the next day when i return from Mount koya and then check in to osaka hotel. i do not really want to being my whole luggage to Mount koya as well. the person might not accept this as well. please advise what i can do about this arrangement or any adjustment i can make?

Thanks!

Many of the major train stations have luggage storage lockers and/or rooms.

https://www.osakastation.com/osaka-station-lockers-and-luggage-storage...

You might have to pay for 2 days if you leave them overnight, but it looks like 700 JPY per day for a large bag, so it's not unreasonable.

I stayed at hotel Kanade in Osaka and hotel M's plus. Both were good value at good locations but I don't know your definition of budget. M's plus was expensive when visited due to Sakura season but should be reasonable during other times of the year.

We travelled to Koya from Osaka and left our luggage at the hotel. Would try to ask for that first and if they won't keep it for you there are indeed lockers at the bigger metro stations like Namba.