Spoiler free text solution, first video contains a very minor spoiler, second is the final battle!For this achievement, you are not allowed to buy anything from the "Dollar Bill" vending machine; don't worry about touching it or looking at it's contents, just don't purchase anything. Use the Gear upgrade or Vigor upgrade vending machines as much as you want.
1999 Mode can be unlocked at the main menu by using the Konami code (up, up, down, down, left, right, left, right, b, a). In 1999 Mode, there is no navigation aid and you have to pay for deaths (100 Silver Eagles). Dying with no money kicks you to the menu, which is basically the same as restarting your checkpoint. If you've finished Brass Balls, then this should be a walk in the park. Playing through first on normal would definitely assist but is not necessary, and you could potentially accomplish all other achievements in this play-through by following the guide for "Infused with Greatness" or purchasing the Season Pass.You can reload checkpoints after the game to get the counter achievements ("Kill N enemies with X"), as they're accumulative across play-throughs.
In case you're curious: without a guide or cheating, I earned around $10,000 (less than a quarter of that spent on revives), and ended the game with an extra $1,000 and 15+ lockpicks in stow. Despite opening every lock I could find, I only found 77/80 of the voxophones and must have missed quite a few other secrets.My Strategy
You're going to have limited resources in this mode, meaning you won't be able to buy much. I chose to mostly upgrade the common weapons (Pistol, Shotgun, Machine Gun, Carbine) Ammo and Damage, and I spent some money on the crucial Charge and Bucking Bronco Vigor upgrades and a single upgrade for the Shock Jockey and Undertow Vigors.
My general strategy was pretty simple: take cover, be patient, and search high and low for supplies. I invested all upgrades into my shield or salts until they were full, and mostly used Vigor combos like shock-shoot to increase damage output on tough enemies. When I encountered a Handyman I would try to always be moving by shooting him a few times and then hopping on a rail to find a hiding spot to regenerate my shield, then rinse and repeat.
For bosses like the ghost, I employed a melee strategy to conserve ammo using the Charge/Charge Aid Vigor coupled with some gear I found laying around: Burning Halo, Executioner, Overkill, and Brittle-Skinned. These are all just random melee perks I had, so if you don't have the exact same perks just pick some similar gear -- stacking melee gear will make your attacks brutal and help conserve ammo. I've attached a video of the fight for you to witness and recreate. After devising this tactic, I used this gear setup for the rest of the game to great effect... Being invincible with Charge Aid is insanely useful even in, or maybe especially in, large crowds. Add Bronco or Shock to the mix and you'll be unstoppable.Finding Gear
Gear always appears in the same locations, but what the gear is is random (except for a few static pieces) so you will have to make do with what you find! If you find a gear location near a checkpoint, sometimes reloading the checkpoint will randomize the gear again. The specific equipment I mentioned is not necessary, I merely chose the melee perk gear I had to boost my damage output.
When combined with Charge and Charge Aid, extra melee damage added to brief invulnerability will help you to cut down heavy hitting enemies with ease. If you don't have the exact equipment, just choose some melee-oriented gear (extra damage is key). Supposedly Burning Halo is always in the beginning behind the first Crow enemy.Choke Point Videos
These videos contain me playing through the two choke points: the first is the fight with the ghost, in which I use a melee strategy to obliterate her; the second is the final battle sequence and contains some pretty heavy spoilers.A note aboute the final mission: I didn't think to do it, but you can set 'Return to Sender' traps in front of the blue generator thing to absorb bullets and prolong the life of the generator.