Plumber issues ... what would be the best way to handle this

And that has what to do with your foolish claim that unless the plumber who botched the repair is allowed to fix the water damaged kitchen cabinets in the neighbors apartment, they have no recourse? Or that allowing the plumber to fix cabinets is the sensible thing for anyone to do? If the lawn maintenance company mowing your neighbor's lawn hit a rock and sent it through your house window, would you have to let some guy from the lawn company fix it? Or would you just get it fixed by a competent company of YOUR choice and then send the bill to the neighbor and the lawn company?

It's very likely that the OP as well as the apartment owners below have insurance. Their insurance companies should cover the cost of most of the repairs. And I doubt the insurance companies are gonna say, "Gee, just call the plumber over and have him fix all the kitchen cabinets and water damage. That's how these things are handled."