Bottle caps can be found while fishing without too much trouble. They are not very common, but are easier to find than the 1% encounter if the fishing spot is not bubbling.
They can also be obtained in the Festival Plaza. Lottery Shops called "Treasure Hunt" have Bottle Caps as the second prize, and the more effective Gold Bottle Cap as the first prize. Whenever a Lottery Shop of two stars or higher is installed, going to the shop immediately and drawing a ticket will always result in getting the second prize, so that's a Bottle Cap from each Treasure Hunt with at least two stars that gets installed, even if it is installed by being introduced.
All Haunted House facilities can also give the player a Bottle Cap, but they are rare.
Gold Bottle Caps can rarely be found from Isle Aphun, found in the Path for Rare-Treasure Hunting. The Path for Interesting-Item Hunting can also give them if Isle Aphun is level 3, as it simply awards the player with items from all unlocked paths. Shards can be obtained from the first path unlocked on Isle Aphun, and 30 of the same colour can be exchanged for a single Bottle Cap by talking to an old man in the castle in the Festival Plaza.
If you go to the Battle Tree, then achieve a streak of at least 30, if you lose a battle and the streak ends the receptionist will give you a single Bottle Cap if you return on a later day. This can be repeated. If the streak went for 50, you get an Ability Capsule, too. PP Ups and PP Maxes can also be obtained from here, and this is how you get the Lansat and Starf berries, with the traditional streaks of 100 and 200 required.
Finally, if you go to Poni Gauntlet and battle all the other trainers there, you can fight Trial Captain Mina. When you defeat her, she will give you a single, once-off Bottle Cap, while the Magearna obtained by scanning the QR code (released) will be holding a Bottle Cap when you get it. If you want to choose the Nature of this Magearna, Synchronize affects it, and works 100% of the time to set the Nature.
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At the end of Melemele Island, you will end up fighting the first Kahuna. After the battle, he gives you the Tauros pager. If you hold B while riding the Tauros and run into the sturdy rocks, they will break.
For sturdy-looking rocks in the water, you will need the Sharpedo Ride Pager, which you get from Grimsley on Route 15. Just like with Tauros on land, charging with B will break sturdy-lookings rocks in the water.