The arrow of time is a concept that has been around for a while. In modern cosmology, it has been argued that there may be different arrows of time in other universes. Since these are also considered to be inaccessible, is it physically meaningful to talk of the arrow of time in other universes?
every universe that qualitatively resembles ours - which certainly includes any universe in a hypothetical "multiverse" where ours may belong - has to have a logical arrow of time. The Universe must know which slices are "input" and which of them are "output". Proofs analogous to Boltzmann's H-theorem then guarantee that the universes also have a thermodynamic arrow of time and the two arrows coincide.
All universes that are in a direct or indirect causal relationship with ours have to have the "same" arrow of time as ours - one that doesn't contradict our arrow of time. It must be possible to define the arrow "uniformly" across the whole spacetime and all of its regions. In particular, one can never mix both arrows of time within the same universe - or the same multiverse. That would be logically equivalent to having closed time-like curves - which are inevitably inconsistencies - because you could get to the future along one arrow and back to the past - which would still be to the future according to another arrow of time.
Evolution by the standard laws, starting with any well-defined slice, automatically preserves the uniformity of the arrows of time even when one allows tunneling and eternal inflation.
The possibilities depends on how diverse the alternative universe can be. Some alternatives may have two or more macroscopic time dimensions, or they may be just euclidean spaces with no time dimension. In that case the concept of arrow of time will not make any sense in those universes.
In other universes which have one time dimension like ours there may be a fixed arrow of time as we (appear to) have. It would make no sense to ask whether it pointed in the same or different direction to ours if there is no direct connection with our universe.
It is also possible that a universe with one time dimension has no discernible arrow of time because everything is perpetually in a high entropy state. One theory is that this would be the natural state in our universe except that for some as yet unexplained reason the big bang singularity had a low entropy that creates an arrow for time. I agree with this view but it is not proven and I don't think there is a consensus.
It may also be possible that a universe has different directions for the arrow of time at different times and places so long as it does not switch directly from forward to backward causing paradoxes. It can switch from backwards to forwards with a low entropy epoch at the changeover, or from forwards to backwards if there is a heat-death epoch between them with maximal entropy that prevents useful information passing.
We don't know the laws of physics well enough to say which of these possibilities can be ruled out on theoretical grounds so they currently stand as just possibilities. There is also a valid philosophical question about whether such questions make sense if we can't observe these universes. Personally I think it is valid to consider them if we have theories that predict the existence of other univserses.
If it is physically meaningful to speak of "other universes", then yes it is meaningful to speak of an arrow of time (and any other constructions in physics) in those universes. Taking this logic further, one could think of doing statistical mechanics over an ensemble of universes each with its own arrow of time - sort of like a spin-system - in the hopes that a "common" arrow of time could emerge. This is in spirit with how ferromagnetism emerges in spin-systems. Even though each spin considered individually is randomly oriented, on treating the many-body system we find collective behavior manifest itself in the form of a spontaneous, non-zero, macroscopic magnetization.
Feel free to criticize this answer for its speculative aspects. However, the question itself is in fairly speculative territory to begin with.