Rival (Route 103): Tackle spam is all you need. Depending on your level, though, you may or may not be outsped and/or need a Potion to win the battle. Carry one just in case.
Gym #1 - Roxanne (Rustboro City, Rock-type): Step 1: Spam Water Gun. Step 2: Win.
Gym #2 - Brawly (Dewford Town, Fighting-type): Marshtomp is generally bulky enough to go toe-to-toe with Machop, but Makuhita will Bulk Up and pull Arm Thrust when you've lowered its health enough. Start off with Mud Shot to lower its Speed, then tread lightly and potentially proceed with Water Gun, to avoid losing damage output due to Bulk Up.
Rival (Route 110): Numel is OHKOed by Water Gun and Wailmer can be Mud Shotted to death. Don't fight Grovyle for obvious reasons.
Wally (Mauville City): Lol, he only has a Ralts. Mud Shot it.
Gym #3 - Wattson (Mauville City, Electric-type): Just spam Mud Shot and watch everything die.
Tabitha (Mt. Chimney): Again, spam Mud Shot and watch everything die.
Archie (Mt. Chimney): Mightyena's Intimidate is annoying but you could probably take it on with Water Gun. Golbat is annoying as well, watch out for its confusion shenanigans. Sharpedo will probably go down to two Mud Shots but Crunch will hurt.
Gym #4 - Flannery (Lavaridge City, Fire-type): Marshtomp's Water Gun/Mud Shot spam will deal with the Slugmas easily. Torkoal is another story because Attract (if your Marshtomp is male, which is most likely) will be annoying and paralysis from Body Slam will be annoying. Also, your Water Gun and Mud Shot won't be doing a ton of damage. It's not a bad idea but it's a difficult battle nonetheless. Bring a Red Flute from the glass shop if you're vulnerable to Attract, too, because you will need it.
Gym #5 - Norman (Petalburg City, Normal-type): Marshtomp's bulk may be helpful here, but remember that no matter what you're going to be taking a lot of damage from the Slakings while doing next-to-nothing in return. Stay away unless evolved, which requires a good deal of overgrinding. Vigoroth is okay.
Rival (Route 119): Same things as before, except now you have a new toy in the form of Surf.
Gym #6 - Winona (Fortree City, Flying-type): If you went and picked up Ice Beam, this battle is so much easier. You should have a Swampert by now too, unless you're running with a level limit. Skarmory will probably fall to two or three Surfs. Pelipper is a bit annoying and is probably too bulky without any weaknesses to any moves you could possibly have, barring the weak Rock Tomb, but Ice Beam still does decent damage. You have the Thunderbolt TM, use it. Altaria is really bulky but is doubly weak to Ice Beam so go right ahead, just watch out for Dragon Dance.
Rival (Lilycove City): Same old, same old. Except there's a Swellow (go ahead with Ice Beam).
Matt (Lilycove Hideout): Just spam your best move, which is Mud Shot in Sapphire. Watch out for Mightyena's Swagger in all cases, it's never fun hitting yourself with that high an Attack, but you should outspeed anyway.
Gym #7 - Tate and Liza (Mossdeep City, Psychic-type): Swampert excels in this battle thanks to Surf hitting everyone super-effectively. If Sunny Day gets up, switch Swampert out of there because not only will Surf be significantly weaker, the Solrock has SolarBeam which Swampert is capable of surviving but shouldn't be attempted. Otherwise Surf spam is the key to victory
Shelly (Seafloor Cavern): You guess it: more Swagger play. Sharpedo is relatively harmless, though, unless the latter Screeches you and prepares the ground for Mightyena's Take Down to hit much harder than it otherwise would. Watch out for deadly combos, but otherwise not dangerous.
Archie (Seafloor Cavern): Always the same. Just Surf Mightyena and then Ice Beam Crobat, use Mud Shot for Sharpedo, or Earthquake if you have it. Yet again, Mightyena's Swagger may be a problem if it outspeeds you, but you should be at a high enough level to avoid this.
Kyogre (Cave of Origin, Sapphire only): Kyogre is at a considerable advantage against Swampert, which can't do a lot to it even with Earthquake, on top of being neutral to Water moves and thus taking considerable damage from them under the rain. Put up the sun if you can, and heal plentifully until the titan is down.
Gym #8 - Wallace (Sootopolis City, Water-type): This is a rather difficult battle for Swampert because everything resists Surf and Ice Beam while being neutral to Earthquake. Probably not the best fight, but you can set up X items on Luvdisc if you wish, and then hammer down the rest. Even if you don't, Swampert's Earthquake is still good and you shouldn't meet many threatening enemies here.
Wally (Victory Road): Altaria can be Ice Beamed. Magneton is ripped to shreds by Earthquake and Delcatty falls to either Surf or Earthquake. Roselia will destroy you and Gardevoir is powerful but will likely be 2HKO'd by Earthquake.
Elite Four Sidney (Ever Grande City, Dark-type): Mightyena and Absol can be Surfed or Earthquaked. Don't attempt to fight Cacturne, even if you're packing Ice Beam (which you really should be). Shiftry has no Grass moves instead, so you can go ahead.
Elite Four Phoebe (Ever Grande City, Ghost-type): You can probably 2HKO or 3HKO the Dusclopses with Surf or Earthquake, but remember that they have Pressure. Banettes and Sableyes will likely fall to Surf or Earthquake but beware of Grudge and Will-O-Wisp, as well as potential Curse damage from the first Dusclops.
Elite Four Glacia (Ever Grande City, Ice-type): Sealeos can be Earthquaked while Glalies can be Surfed or Earthquaked. Walrein is too bulky and can stall you out using Hail damage and can even pull Sheer Cold, which you can only avoid by over-leveling it. Use something that hits it super-effectively if you can, otherwise be ready to heal.
Elite Four Drake (Ever Grande City, Dragon-type): How to own Drake: spam Ice Beam. That's it.
Champion Steven (Ever Grande City, Steel-type): Skarmory falls to Surf-spam while Aggron falls to either Surf or Earthquake. Metagross is 2HKO'd at worst by Earthquake. Armaldo and Claydol are killed by Surf. Don't try Cradily due to Giga Drain.
Post-Game: Does anyone even nuzlocke it?
Yeah, Mudkip is the best starter.
Source: https://project-nuzlockedex.fandom.com/wiki/Mudkip_Line/RSE
