Hi, I'm Cosmic
I'm an Achievement Hunter/Completionist and I saw that not many people have this achievement so I thought I would help people who are stuck.

So far I've found two options on how to get this achievement.
1. A Difficult Grind
2. Mods

I'll cover how to do both in this guide

In the pub in town there is an Arcade Machine for the game 'Journey of the Prairie King'.
To get this achievement you have to beat it without dying

This can be very difficult, mostly relying on being lucky with coins and power-ups.
Beating each level shouldn't be too much of a struggle but doing it no hit is mostly a test of perserverance, skill and luck.

Outlaws are pretty easy to kill no hit, just keep your distance and avoid their bullets, which should be easy if you've played Bullet-hell games before.
With Fector, it is easiest if you have either the Sherrif Star or the Machine Gun when you start it.
He starts shooting instantly, so the best strategy is activate the power-up (sherrif star or machine gun) stand in the centre and shoot down at him. He normally wont last long if you do this

Here's a video example: (Fector is at about 19:20)

Modding the game isn't too difficult.
It's as simple as downloading SMAPI and running it: https://smapi.io/
Once installed, you can find mods online on websites such as Nexus Mods.
Here's one for The Prairie King that i found: https://www.nexusmods.com/stardewvalley/mods/5816

There should be a guide for setting up SMAPI but to make sure achievements work, after the string SMAPI asks you to put in Launch Options, make sure to put %command% at the end of it so that achievements work with mods.
Example: "C:\SteamLibrary\steamapps\common\Stardew Valley\StardewModdingAPI.exe" %command%

I don't think this is how it should be done however it makes it easier for those who don't want the challenge

Credit: josh

Play the arcade game up to stage 2-2 (not difficult with good RNG)
Leave the arcade game and sleep a day

You can then go back to the game and keep trying to advance to Fector

Each time you die you can just reload your save file to start the day again

Thanks to TheK9Master :

Here's a spot where Fector can't hit you (potentially bug or is just how his attacks work?)
https://gtm.steamproxy.vip/ugc/1820015997355562836/F507C8029F44C32825B9B50B0E4076BB9B0E3155/?imw=256&&ima=fit&impolicy=Letterbox&imcolor=%23000000&letterbox=false

Thankyou for reading my guide,
I hope it was helpful.

Please leave feedback in the comments, and rate it if you found it useful
~ Cosmic