"Complete a giant-robot boxing match without taking damage". There's almost no information online about this so I'm just going to put this here. This is by far the easiest way to get this very annoying-to-get bottle-cap.
Guide to Getting Bottle-Cap
What You Need
- Thumbs/fingers to mash with.
- ~5 minutes of time.
- Operation 005-C unlocked.
- Difficulty set to Very Easy.
How to Do It
Reach the Wallgah-Goojin boxing fight in Operation 005-C, with difficulty set to Very Easy. This means doing the entire Gah-Goojin boss fight and the lead-up before this punch-out section. If you somehow get hit during the Wallgah-Goojin fight you need to do everything before it because there's no checkpoint at all (thanks Kamiya).
Also just skip all the cutscenes. It also doesn't matter if you get hit before the Wallgah-Goojin fight so don't fret if that happens.
As soon as you get into the fight just start mashing the punch buttons, alternating the two punch buttons (default punching). He will block every punch you throw, but this is what we want to happen. Do not bother with the KO Punch button. He will also never counter with a headbutt because you're playing on Very Easy.
After punching a bit just start holding Up on whatever device you're using to start doing high punches that'll aim for his head. If you keep switching between high and low punches (not rapidly, just every three or four punches), one of your punches will go through his guard randomly and damage him. After you see one of them damage him, let go of Up and you'll start doing low punches again. Punch a couple times this way, then switch to high punches again. There is no strict timing for switching between low and high punches so don't get worried. You can just do it faster if you switch punch types when you see him get damaged.
Rinse and repeat punching low and high until he is inevitably knocked over.
When he gets up, and you can continue punching the same way as before until he runs out of health.
Don't forget to do the QTE successfully when he runs out of health.
Right after this QTE, the bottle-cap should unlock if you did it right.
Alternate between some low punches and some high punches infinitely, he will block most of them and never counterattack since it's Very Easy mode. He will quickly die after 2-3 minutes of punching, granting you this achievement as long as you never took damage.
Now you can get Wonder-Goggles a little easier than before, without having to worry about this incredibly annoying bottle-cap.