A “Universe Today” article titled “Galaxies Are Moving Away From Us Faster than the Speed of Light” prompted the following question, which the author of the article was unable to answer:
“Would such galaxies, relative to ourselves, be moving backwards in time?”
Questions like this are complicated because you have to be clear what you mean by time. The simplest definition is that time is what is shown on a clock, so if I was holding some hypothetical clock that had been reset to zero at the Big Bang my clock would currently be showing 13.799 billion years i.e. the age of the universe. The question then becomes what does the clock held by an alien on a distant galaxy show?
The answer is that (to a good approximation) every clock holding alien everywhere in the universe, even in galaxies moving away faster than light, has their clock showing the same time of 13.799 billion years. So not only is time not running backwards in very distant galaxies, but it's running at exactly the same rate as our time.
Explaining exactly why this is the case requires going deeper into general relativity than I suspect you'd wish to. You may have heard it said that distant galaxies aren't really moving away from us, it's just that the space between us and the distant galaxy is expanding. What this means technically is that we describe the geometry of the universe using comoving coordinates, and in these coordinates the distant galaxies aren't moving. Because they aren't moving there is no time dilation so they all agree on the time. We call this time comoving time and it has the property that it matches the time shown on a clock, just as I described at the beginning of this answer.
If you're curious to know more about this have a look at my answer to Doesn't dating the universe violate the concept of spacetime's inseparability? as this discusses the subject in a bit more detail. I'm afraid that to really understand the problem will require you to learn general relativity.
This is true for galaxies beyond the cosmologic horizon.
BTW they are not moving at that speed: their apparent speed seen from our place is such. Quite like the fact that the far galaxies we see close to the horizon seems both very young (which they aren't "in real life"), very red-shifted (while their emitted colors are indeed normal) and living very slowly if we could see (their time seems frozen for us, but not for them).
So you might say that beyond horizon the "apparent time of now definitively invisible galaxies" goes backward if you really want, it has no real impact for anybody :-)
