This is a collection of helpful tips that got me through the campaign on 1999 Mode. I'm not very good at FPS's and this was definitely a challenge, but it's very attainable.
*Once you get your hands on a sniper rifle, NEVER LET IT GO. You are not durable in this mode at all, so anything that can keep the fight at a distance is almost always your friend, and few things keep the fight at a distance better than the sniper rifle.
*Possession should be your go-to vigor. Sure, it eats up a lot of salts, but it's a guaranteed kill on everything other than machinery, heavy hitters, and zealots, and there are always plenty of salt vials around. You can just toss a possession ghost out there, and as long as there's a clear line of sight to your target, it will usually track them down. No lining up your shot, just get it in the right general area and retreat to cover. Try to possess snipers, guys with flak cannons, and guys with RPGs if possible due to the enormous amount of damage they can do to their comrades instead of you. Purchase "Possession for less" at a Vini Vidi Vigor machine as soon as you can.
*Pull up a collectible guide and track down every infusion you can. Once you've maxed your health and salts, shield is not as high priority since the increase is so marginal anyway, but every little bit helps, and you should be searching everywhere for money, health, and salts anyway.
*Alternate your infusions early on between health and salts until you've achieved enough salts to get one extra use of Possession. Then max out your health, salts, and shield in that order.
*Only spend money to upgrade weapons and vigors you use, and be very selective at that. For instance, if the recoil of a weapon doesn't bother you much because you tend to shoot fairly slowly anyway, don't waste your money. Don't plan to use Undertow for multiple enemies? Don't buy that upgrade. Hoard all of the rest of your money because you'll need it for respawns.
*Don't waste anything. If you only have about a quarter of your salts missing, don't pick up a +50/+100 salt vial just to top off. Same goes for health. Wait until you can use every point before you pick it up, unless you're about to leave an area for the final time, then go nuts.
*Bucking Bronco/Undertow makes for a good combination if you'd rather just send crows and firemen flying off the edge of Columbia instead of having a prolonged battle.
-I personally did not use Murder of Crows or Devil's Kiss once.
-I occasionally used Bucking Bronco and Shock Jockey to immobilize Crows, Firemen, and Patriots (only Shock Jockey) but otherwise only rarely used them.
-Undertow can be helpful for restraining Handymen or knocking enemies off Columbia (which is not an option you get often).
-Charge is super helpful for the Siren but I never used it otherwise.
-Return to Sender is great for patriots and the final battle and was probably my #2 vigor.
-As mentioned above, Possession was my go-to at all times. Its usefulness cannot be overstated.
- You can skip the first handyman fight. Just run to the club from the gun shop as quick as you can. When you return to this area, he'll be gone. Maybe you're a better crack shot or strategist than I am and you can handle the fight, but for me it just wasn't worth it.
- As far as I can remember, every other handyman fight occurs with a skyline in the area, so the gear combo of Winter Vest and Firebird is your friend. Just keep jumping on and off that sky line like a madman. Maybe take a few shots at his heart during those few seconds of invulnerability, but don't be a hero. It'll be slow and tedious, but keep spamming that sky line and you'll eventually wear him down. Gear is randomly generated, so in order to obtain those two pieces (Winter Vest should be your priority), find a gear box that is relatively close to an autosave point. I say this because if you don't get the gear you want, you can reload your checkpoint and try again. Not a great strategy if you have to re-do significant portions of the game because of the distance between the checkpoint and the gear box, obviously.
*Patriots: Your first few patriot fights are gonna be rough and hard-fought. Sometimes shotgun blasts can knock them off balance and you have a chance to rattle off several shots before they regain their composure, but it's going to be a lot of Shock Jockey followed by running around behind them to attack their gears. Alternatively, if you have a high enough powered mid/long range weapon (hand cannon, sniper rifle, RPG, volley gun, flak cannon), forget the gears, just go to town and dump several clips into their front. They might not take as much damage as they would to the gears, but you'll be much safer at a distance.
*Once you get the Return to Sender vigor, you may officially declare yourself the grim reaper of patriots. Make sure your salts are topped off, put up that shield, stand right in front of them, let them unload, and as soon as they pause, hit them with that ball of pain and anger you've been building up as they've unloaded those crank guns on you.
*Don't bother possessing patriots unless you just need a few seconds to make an escape. The effect usually wears off before they've even had time to get their gun going.
*Lady Comstock/Siren: Credit to NightmareNignt over at GameFAQs for this one. This works for all 3 fights.
-GEAR: Make sure you have the following gear, and these are essential: Brittle-skinned, Burning Halo, Pyromaniac. For your fourth gear slot, anything that upgrades your melee damage is good but not needed. I went with Overkill (which is not melee-specific but whatever) and it worked great.
-VIGORS: Charge. This strategy works perfectly fine with a completely non-upgraded Charge vigor, but any upgrades you can throw at it will make it even more effective and helpful.
-WEAPONS: I personally didn't need to even use my weapons, but the Hailfire is recommended by NightmareNignt.
-STRATEGY: Make sure you have those gear equipped and your salts are topped off, and just go to town on her with Charge. I finished all 3 stages of the fight in under 10 seconds each. She seems especially vulnerable to fire, which also takes out her minions, and with brittle-skinned making her extra vulnerable while charge makes you invincible, this combo just absolutely demolishes her. Like I said, I never even had to use my weapon.
*Final battle: Credit to... many people. Hard to track down who exactly originated this strategy so I will just say that many people have posted it across various forums, and it's not mine originally. So I will just describe my personal approach, which incorporated many things from many people.
-Surround the core with Return to Sender traps. These traps will absorb probably 80-95% of the shots taken, and as they absorb more shots, they grow and fill in the gaps. Pretty soon it will only take one very occasional stray bullet. Make sure you surround the entire core, not just the half facing the bow of the airship as enemies will come up behind it and shoot from there.
-Explosives still damage the core, so possess/kill any RPG/flak cannon/hailfire enemies ASAP.
-Periodically check on your traps. I didn't notice how or when it happened, but some of the traps were periodically triggered and would leave a gap in the defenses. Fill the gaps back up with traps and resume what you were doing.
-Use Songbird immediately when you can. Have him take out the troops on the deck FIRST, and THEN the gunboats or zeppelins. This seems counter-intuitive because of how much damage the larger vessels can do relative to the foot soldiers, but Songbird recharges from his attacks on the deck very quickly. So rather than take out the zeppelins and letting the foot soldiers go to town on the core for 2-3 minutes, take them out first, and then have Songbird attack the large ships after 15-20 seconds or so. This way, rather than letting Songbird just sit there unused during the time between waves because his most recent attack was on foot soldiers, you can have all of that downtime count toward his cool-down period. This is really just a matter of preference for you though. If you keep your traps in place and make sure you get rid of the guys wielding rockets/grenades quickly, you'll have plenty of health in the core left at the end no matter how you use Songbird.