Here is how I got this on my first attempt.
All it says is "solve all case files in co-op". In other words, you only need a second player when a case is "solved" and the next case is assigned. They don't even need to be there when you give Zombrex to Katey. If you aren't about to solve a case, you don't need a co-op partner at the moment.
Host the game. Partly because it's hard to find an online game where the player is actively trying to solve cases, but mainly because you want to control when the cases are about to be solved. Whenever you get a co-op partner, work on cases (obviously) but when you lose your partner, save your game to lock in the progress you've made, and this is very important ...
Prepare for the next case (food and weapons)
Go to the closest save point to the next case
Let time pass until the next case can be worked or is about to start, and save the game
If no one joins and the clock is about to run out on the case, re-load from the last save
At one point, the next case is 12 in-game hours (over an hour in real-time) away and my co-op partner left, so I played alone almost until the next case and then made a convenient save. I re-loaded the save many, many times waiting for a co-op partner.
To prepare for the next case, remember that your co-op partner may spawn with no food or weapons, and might "suck at killing zombies" (to quote Leon). You'll need a lot of equipment slots so the higher your level, the better. Make some Painkiller potions, or at least have foods with high health restorative value. Don't think of your partner as a helper; think of your partner as a liability who needs to be kept alive.
To fight Sgt. Boykin, you must go from the Safe House to an underground area with gas zombies roaming about. One of my partners couldn't even get through the gas zombies so after he left the game, I went to the closest savepoint to Boykin (underground restroom you pass in Case 2-2 Ticket to Ride immediately before you fight the 3 mercenaries and then chase the train). When you re-load, you need to get out of that area pretty quickly (perhaps by dashing to the motorcycle) but just go to the large area right outside the Boykins fight, kill the dozen regular zombies and wait for someone to join.
Find all of the "free" Zombrex as soon as you can. It is hard enough solving cases in co-op with random players. You don't need the extra burden of being forced to do the side missions to keep Katey alive.
There are comments on whether to save when the game offers after a case is solved, or to save manually. I did both types of saves and got the achievement, but I can't say for sure which was the last type of save I did whenever I quit the game and re-started later.
I don't think I delivered the last shot on the final rooftop boss, but still got the achievement.
All of my saves were to the hard drive. At first, I wanted to play with a particular friend but 99 times out of 100 we would get "error while trying to join" no matter what combination of skills packs, USB memory sticks, and old Xbox Memory Units we tried. Eventually we gave up and I played with randoms and always saved to the local hard drive.
Much of what I've written is sprinkled in the comments in other solutions, but so is a lot of contradictory information. I wrote this as a standalone solution so that it would be a single, coherent explanation.