If you synchronize the pushes at the two ends then you can make the ends stop at the same time. Or you can make one step before the other, either way around.
Consider the following related idea. Suppose New Year's day is approaching and we arrange firework displays all around the world. Everyone has an accurate clock. We can arrange the start of the displays so that a sequence of rocket launch events happens in all the cities, one just before the next, and with careful preparation the sequence can proceed faster than the speed of light around the world! Or to have an example involving longer times, put the events on different planets. Then the timekeeping will not have to be so precise. It will be pretty easy to arrange for an event on Mars to start just a few seconds after an event on Earth for example. But you don't do it by sending signals from Earth to Mars just when the Earth event happens. Rather, you plan ahead and a day or so before you could tell the people on Mars to get ready and then do it at the agreed time.
The point is that by preparation one can make things at different places happen simultaneously, or one after another, or one before another, however we like. But the preparation signals---the ones used to set up the experiment---will all have to travel slower than light.
(If with the long rod you just stop one end, then an impulse will travel down the rod at the speed of sound and eventually the other end will stop, but that was not the experiment you had in mind.)