Yes. As others have said, it will require really extensive prefabbing. I propose a somewhat different approach to building it, though.
Build four "buildings" around the final job site. In each building you assemble entire floors (one has floors 1, 5, 9, 13..., the next has 2, 6, 10, 14... etc.) The buildings have beams connected across the final jobsite.
When the clock strikes midnight you slide the floors over to the final building, fasten all the beams together and pour your quick-setting concrete. At 23:59 you fire the shaped charges that will cut the connecting beams, then fire the charges to bring down the outer supports of the surrounding buildings. While a building of that size won't truly topple it's going to tip outward as it falls--you can get an awful lot of directionality to the debris field. Combine that with separate partition walls between the buildings and the final structure to corral stray bits from the demolition and you should be able to pull it off.
Now, the organization of the steel in the resulting building won't be the same as the original but overall it will look basically the same.
(I do, however, worry about blowing windows when the cutting charges separate the beams used to move the pieces into place.)
Unlike the other proposals this allows moving every floor into place simultaneously and I don't believe any possible solution exists that doesn't do it basically simultaneously.