If you are interested in playing through the game on hardcore, I recommend ignoring this guide and reading the one provided by Silverblade. This guide assumes you're only playing Hardcore for the achievement, nothing more. That said, you can get the achievement by the third in-game day, though you'll be taking some huge risks. There are exactly three things you need to do for this, and it shouldn't take any longer than an hour, compared to a proper playthrough which should take days at the very least. You also should, assuming you're fast about it, be able to complete this before the real threat of hardcore (hunger, thirst and sleep) actually becomes a threat to your continued survival.
1) Character creation - Put all your points into agility, and tag sneak, as well as your preferred combat skill. I'd recommend either guns, as you'll be siding with the NCR for the final battle and can scavenge from your fallen allies, or melee, so you can scavenge from fallen legionnaires.
2) The path to Vegas - If you're new to New Vegas, immediately quit following this guide and get used to the game, because you'll be taking the Quarry Junction shortcut, also known as "suicide by deathclaw" for the unprepared. This is the only section that can be skipped, and only if you don't think you can survive the shortcut. Immediately after leaving Doc Mitchel's, head north through Quarry Junction as quickly as you can, doing everything you can to stay away from the deathclaws. If one so much as touches you, you'll most likely die, and even if you live, you'll probably be crippled anyway, which is a death sentence on hardcore, especially when you're so underprepared. There is another path, also to the north, just up from the Goodsprings gas station, but be warned that this path has cazadors (faster than deathclaws and with poisonous attacks, and unless you've got the preorder DLCs, you won't have any antivenom for that poison). Just do everything you can to get to Camp McCarran, ignoring any Fiends or beasts on the way.
3) The Second Battle of Hoover Dam - For this, we're doing the laziest form of the Yes-Man path. Immediately upon reaching the ruins of Vegas, sneak into the New Vegas strip via the Camp McCarran monorail, then head straight to the Tops, up to Benny's room, and speak to Yes-Man. Then head to the Lucky 38 and assassinate Mr. House. With that done, return to Yes-Man, return with him to the Lucky 38, then ignore every side faction. You'll then have to head to the Dam for Kimball's speech - you don't actually have to save him from any of his assassins, just make sure he lands, then leave him to die if you want. Return to the Lucky 38, start the final battle, then rush to the Legate's Camp ASAP, avoiding combat where you can and gathering ammo and weapons from dead NCR and Legion troops. When you reach Legate Lanius, kill him (reminder: Hardcore in New Vegas is not Survival in Fallout 4, so feel free to set the difficulty to Very Easy if you want), then retreat to the camp entrance. General Oliver's rangers will mop up the remaining legionnaires for you. All you have to do now is decide how Yes-Man should deal with the "good" general. Once peace has been negotiated or Oliver has gone the way of his Legion opposite, the credits will role and you're done.