As has been said, this isn't overly more difficult than the original hardest difficulty, Nightmare.
A few tips, really, are all you need to succeed, as the game does a good job of helping you along.
-Stay in cover and try to shoot enemies through gaps in that covering (pipes, cracks in the wall, etc.)
-Go for headshots when you can, to conserve ammunition and make the enemies go down faster.
-As far as what weapons to use, I found the Fists of Rage (extra from the Anarchy edition, thanks to bytyqi for bringing it up!) to be helpful in conserving ammo at the beginning of the game, especially against the Mutants who don't shoot at you.
-I also really liked using the assault rifle (with feltrite rounds) and the authority MG (with AV2X rounds). I found that the regular ammunition wasn't even close in comparison to the effectiveness of the feltrite and AV2X rounds, especially towards the end of the game.
-The nailgun is also hugely helpful (thanks MCoreBE for bringing it up!), as the rebar rounds will kill most enemies with one hit, and the railgun slugs can shoot through walls and kill multiple enemies with one shot. However, it will require you to play through the first mission of the Scorchers DLC, which is not required for this achievement.
-Cerichsen also recommends using the Fat Momma ammunition (it will require you to obtain the schematic) for the Pistol and the Pop Rockets for the Shotgun, as well as keeping a few Sentry Bots around when things get sticky. (I preferred to sell everything I didn't absolutely need and it helped me stay stocked up on good ammunition, but you'll have to find out what works best for you).
-(Any of the special ammunition is extremely helpful)
-Try to keep a healthy supply of bandages and don't be stingy with using them - they only run at about $14 in the stores and it can save you a lot of hassle from having to reload/waste your defib over and over.
-The enemies that shoot you are your friends! The only problems I had where the melee enemies that rushed you and the heavys, which just kept on coming. When the enemy takes cover to shoot at you it gives you time to analyze the situation and figure out what you want to do.
-Find a way to make money to buy supplies! I used the Tombstone minigame a lot. SAVE right before you start playing, bet the maximum amount ($25) and try to kill all four before they get to you. If you win on the 10x or 4x round, back out and save again. If you win on the last round that's fine, as you at least keep your money. If you lose, load your saved game and try again. It may not be the fastest way to make money, but it doesn't require any consistent skill to win and it helped me buy the armor upgrades as early as possible.
-Cerichsen said that if you are willing to play through the second Scorchers mission, it will unlock Jackpot in Wellspring and he made between $5000-$7000 from the video poker game in about 30 minutes.
-I didn't bother with any of the side missions or collectibles, as I had already gone through and gotten 100% prior to the release of this DLC, but if you are going for the 100% just remember to be careful and take it slow - it may make you feel like it takes longer than just running through, but all that "time you save" quickly goes into the negative when you end up having to reload to try it 4-5 times.
-This is for the campaign that came shipped with the game - it does NOT require you to do the Scorcher DLC add-ons.
-****This achievement has been confirmed to be stackable with the other difficulty achievements**** (Thanks and congrats to X InFlames80 X)
Please feel free to add comments for any way I can make this better - I'm not putting a full mission by mission walkthrough as those are easily found on the internet, and these were just a few tips that helped me make it through the Ultra-Nightmare mode.
MCoreBE is uploading a FULL playthrough guide of Ultra-Nightmare difficulty. Link below:
http://www.youtube.com/playlist?list=PLFYg0IPAUNeAgybjGzE6HK...