To anyone reading this before starting the game: Fector's Challenge is a very difficult trophy, but don't let that stop you from experiencing Stardew Valley. It is an outstanding game, developed by a single person. So much so that I'm completing it on PS after already doing so on Steam.
I was able to complete this challenge in one day, so I will try to give my tips as concisely as possible. It's best to already be familiar with the mini-game (level layouts, upgrades, etc). The only aspects of the game that are RNG dependent are the spawn rate of coins and the upgrade spawns (especially during 3-3). Everything else will require practice, trust me it gets easier with practice. Keep in mind, these tips are my preferences and are not an absolute guide. Different players will have different preferences.
I have posted my attempt on YouTube at this link: https://youtu.be/4xHVf2cjydg. I completed it on PS Vita. I will be referencing this through my tips.
1. Do NOT go after extra lives (for obvious reasons).
2. Always use your stocked boost to avoid death in a pinch. If you die, the whole run is meaningless, so use every resource you have to avoid death.
3. Prioritize going after coins in lower stages, but don't risk your life in the process.
4. Camping in corners is extremely useful when overwhelmed. Enemies seem to get confused when you are in a corner sometimes, and they move much slower in a diagonal line. In later levels leave enough room in the corner to deal with randomly emerging flying enemies. This can be seen throughout my run in my YouTube video.
5. The first upgrade after 1-2 should always be ammo. This will make 1-3 and 1-4 a cakewalk. It also makes Stage 2 much easier. If you do not have enough coins by this point, just restart. Trust me, its better than realizing 10 minutes down the line that you aren't prepared.
6. The second upgrade after 1-4 should be the gun in my opinion. The fire rate is crucial to surviving the Stage 2 hordes with ease. If you do not have enough coins by this point, just restart.
7. The two sheriff bosses on Stages 1 and 2 can be easily cheesed. Check my YouTube video at 6:40 for Sheriff 1 and 12:15 for Sheriff 2.
8. During the entirety of Stage 2, you need to save up for the second ammo upgrade. Otherwise, it is VERY difficult to survive Stage 3. So, don't purchase anything after 2-1 if you can't afford ammo 2.
9. Stage 2-3 is notorious. My video shows a very easy corner camping method starting at 10:29. Buy ammo 2 after this stage.
11. After 3-1, purchase gun 2. Once you have ammo 2 and gun 2, the rest of stage 3 should be easy enough, including 3-3.
12. As dumb as it is, 3-3 requires good RNG to survive the horde. You just have to get lucky.
12. To cheese Fector, you MUST go into the fight with either a Sheriff's Badge or Machine Gun upgrade. Otherwise, you have to fight him while dodging his fire, which is very difficult.
If you managed to win, congrats! Just keep trying with this strategy and you will win eventually. You just have to wait for that one lucky game. Good luck!