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This isn't actually as difficult as you'd think. I personally found this to be a cake walk when compared to the hardest difficulty in other games, such as CoD or Halo Reach's "Monument..." achievement.
If you work at getting skill shots, and make sure to catch all the reflex scenes as best as you can, you'll have more than enough ammo and upgrades to go through. By the end of the game, I had maxed out every gun and full ammo for everything as well.
Benni Krok has suggested that if you have issues with the train section in Chapter 1 Act 2, to leash the vulture pilots instead of trying to shoot them down. I think this is a good suggestion any time you're up against Vultures, as it conserves ammo, and is much quicker. Unless you have a sniper rifle, which is always just loads of fun to use on anything.
Suggestion on how to defeat Weed Killer easily: Make sure your flail gun & PMC are fully upgraded for capacity. Don't bother buying ammo for either this or your PMC. Ammo for both these weapons will drop by killing the flytraps just before this battle. Third weapon doesn't matter, but I brought the flare just in case. When there is a health bar, use two flails then switch to your PMC to finish off the last little bit. Once it goes upside down, if you are quick enough to take it out, it will not have a chance to drop those annoying puff balls. Use one flail per "glowy part" as Gray says. If you aim it right, you can take out two with one flail. I only did this once, but you don't have to as you should have plenty of ammo if you don't miss at all.
One important thing to note is that after you do "Weed Killer" - DO NOT buy ammo for your guns. I made that mistake. Shortly after that battle, there is an ammo container which will give you full ammo & charge for all the weapons you're carrying. You may want to bring along a weapon you enjoy using as your third for the ammo, since you shouldn't need to use any ammo from it for that battle. There's a second weapon cache underneath the city when you're pinned down by snipers. Make sure to switch through all your weapons and fill them all up here, as it will be very helpful on your cash flow. To aid in this, there's a drop shop right next to the ammo cache.
Several people have also posted issues with the elevator section. The best way I've found to defeat this is using the flail gun. Make sure you have capacity fully upgraded and have filled the gun. Don't bother with the turret, as it takes too long to aim. Look upwards, and fire between the two gears that you are protecting, on the corner of the shaft. Then look down, and detonate the flail as the highest of the crawlers is just about to reach the flail mine. Make sure you detonate it before he gets there, as if you wait until he's right next to it, you will miss him.
A short note on the helicopter section. Don't bother going for the red barrels achievement. That is much better done on Very Easy after you've completed the campaign. Make sure you get past the first part with at least 30% health. The first part is the hardest. After you get past here, you should not take more than 10% damage (if that. Once I managed to pass the initial area, I didn't take any damage). Don't forget to only zoom in when you're shooting at something. Trying to move your camera while zoomed in slows it down a ton.
Macross Soldier says: "Another tip for the helicopter section is that flares can be shot out of the air, so if you're having trouble killing them you can at least keep them from doing too much damage."
The most important thing is kill the boss right away, then kill the two vultures as soon as they show up (before Trishka calls them out, if possible) as they will kill you faster than anything. After that is the flair gunners. Use the red barrels and big gas containers to your advantage. Shoot them out to kill the boss and the flair gunners that run out from the building.
If you manage to kill everything and still have some time where you're hovering at the start, don't just sit there - look up and aim at the second platform and start shooting the barrels and people up there.
The only two spots I had any real problems were with the helicopter section & the "get to the jumpship" section. Hope this helps, good luck!
*Note: This achievement does stack with the other difficulty modes.
*Note 2: You can select each chapter individually on chapter select after completing the game on an easier difficulty, and still get this achievement. You will not, however, have bonus SP if you have not finished the chapter on a difficulty harder than you have previously finished that chapter. (Thanks AntonySJ)