On my normal difficulty playthrough, the only thing that really gave me trouble was the Handymen. They seem to have a counter-attack for every strategy, and it takes a lot of firepower to bring them down, regardless of your weapons.
Now that I’m on 1999 mode, I know that these guys are going to be more or less impossible unless my aim is true and my strategy flawless. What strategies can I employ to take down Handymen more easily?
The most effective vigor to use against Handymen is oddly Murder of Crows. While this vigor does not directly damage the Handymen, it causes them to stop and swat in the air for a couple of seconds, leaving their heart critical open. Playing through on 1999, this was literally the only tactic that allowed me to kill them. In the same way you can perma-stun Zealot of the Lady (the crow enemy) with Shock Jockey, you can almost permalock the Handymen with Murder of Crows.
Also, if you are playing through on 1999, be aware that there is a new chapter before almost every single Handyman, so if you find yourself dying too many times and want to use the lost money for upgrades, it may be better to back out to the menu while dying as the Handymen will recover all of their health when you respawn.