I can see two options with that. One way is to lift it off with a helicopter, based on the budget that building has, that’s probably how they’ll do it. The other option is to dismantle it and lower it down through the center, possibly abandoning the last parts of it on the roof.
On some buildings I’ve seen, that small crane you mentioned (“winch on a short beam attached to the roof somehow”) is used for supporting the platform used by the window washers, and is stored permanently on the roof.
The same could be asked…how does the first crane go up? These things are modular, but what lifts the modular pieces into place to begin with? I assume a helicopter is the only answer.
To answer all your questions at once, that crane, which is a “luffing crane,” is exactly like the ones in the OP and will be taken apart using a big-ass temporary davit arm system put up on the roof of the tower. A davit arm is that “winch on a short beam” that is usually found on building roofs to raise and lower swingstages for the window cleaners. The luffer has a jib that can be raised and lowered to bring the load closer or farther away from the structure, which is slower than a typical tower crane, but much more versatile. When they take it apart, the jib will be lowered to horizontal and dismantled section-by-section by workers using the davit arm to grab and lower the pieces, after which the mast will be taken apart the same way. No parts will be left in the building – these cranes are worth a fortune and just leaving pieces behind is simply not an option.
The helicopter option is used infrequently, for a number of reasons. First, hooking up the parts and releasing them takes a lot of time and hovering the chopper in a completely stationary position for that long costs a ton in fuel. Second, if anything goes wrong – the load becomes unstable, a sudden wind comes up, the pilot really has to pee – the required operating procedure for the pilot is to cut the load free and save the helicopter. And dropping a couple thousand pounds of steel onto a downtown street or neighboring building from a dozen or more stories up results in a TON of follow-up paperwork, which is very inconvenient.
Finally, as I noted above, to initially erect one of these cranes, it is typically done right at ground level, using a mobile crane, to get the tower its initial hundred feet or so of height, after which it is raised, using one of the methods I noted earlier, as they build the structure up around it.
