BOTW Can I save Yunobo in the Abandoned North Mine without the cannon?

No

So, it's possible to complete Divine Beast Vah Rudania without saving Yunobo – this can be trivialized with MJWW (Moon Jump Wrong Warp). However, it is impossible to actually save Yunobo.

If you would like to just complete the Divine Beast and don't care about Yunobo, here is your solution.

Prerequisites

• A Normal Mode file with:
  • A registered horse
  • Divine Beast Vah Rudania incomplete and without the map, but accessible
  • The Travel Medallion
  • At least 20 Rupees

Steps

1. Load up your Normal Mode file.
2. If you've never placed your Travel Medallion, do so. Place it anywhere you like.
3. Warp to Divine Beast Vah Rudania.
4. If the game doesn't autosave, save manually.
5. Go to the Mounted Archery Camp. It is located southwest of Lake Tower, northeast of a small forest called Pappetto Grove. If you don't have the Ancient Saddle on your currently taken out horse, ride your horse there. If you do, you may also choose to teleport your horse there once you're there.
6. Get on your horse, and talk to Jini while riding your horse.
7. Start an archery drill.
8. Upon starting the drill, turn your horse around immediately, and ride it up the small hill just ahead. Get your horse onto the hill at the front, and and ride it along the hill until you see a single bush on your right.
9. Turn your horse to face the bush. There should be a small patch of horses just ahead.
10. When the timer reaches 10 seconds left, jump off your horse and glide towards the patch of horses in the field.
11. Drop onto a non-spotted horse. If done correctly, you should automatically mount it and have the Soothe prompt pop up, with the horse attempting to buck you off.
12. Wait out the rest of the timer. Do not press L while on the horse.
13. You should see a glitchy cutscene, with the registered horse standing at the start and Link riding the wild horse, with the wild horse strangely bouncing around in the air.
14. Quit the minigame. As amusing as it is to watch your horse bounce around, if it gets too far from Jini, the game will softlock.
15. After the dialogue is gone, load your Divine Beast Vah Rudania save.
16. Fast travel to your Travel Medallion.
17. Load your Master Mode file.
18. Save the game, and reload that save.

If everything was done properly, you should now be inside Vah Rudania. Complete it as normal.

However!

This will lead to there being two Vah Rudanias, and glitchy game progression as there is a complete one and an incomplete one. This will not count as progress towards the Vah Rudania quest, but counts as you having completed it in most other aspects.