You asked: "Does one imply another?"
No. Neither implies the other. However, I think there are benefits to first being clear what the ideas are, particularly since I think each idea actually already assumes an arrow of time.
In the first case, you start with an arrow of time that only earlier times affect later times, and then end up strengthening that to be that causes have to be within the past light cone in order to have causes precede effects in all frames. When they say past, they mean past light cone. There is already an idea of cause and effect with cause coming before effect.
In the second cause, again when they say that entropy increases, they already have an idea of earlier times and later times when they say entropy increases. In this cases they only mention that you can measure this direction by looking at entropy measurements.
So, now, does one imply the other? An easy way to show that A does not imply B is to exhibit an example where A is true but B is not. Sometimes the example really just reveals a third thing that was needed (i.e. if when A is true and B is not you realize that C is not true then you might turn around and state that A&C together jointly imply B).
Neither implies the other because the first is a not really operationally defined in a non-tautological fashion.. So let's look at an example where the first is false and the second is true (this will show the second cannot imply the first). The idea of causality presented in the first idea makes no predictions. When you make a theory, you simply predict that observations are restricted to a smaller collection of possibilities, those selected by your theory. The theory is falsified when observations are observed that do not belong to the limited collection. And making the collection be limited is actually what the theory does. For instance Newtonian Physics is a theory about a correspondence between solutions of second order Ordinary Differential Equations and the observed dynamics of material bodies. The correspondence is what the theory is about. The equations themselves don't identify a preferred direction for time, you can tell stories after the fa t and call some thing causes and other things effects, but it is just a story on top of the actual theory and doesn't carry any scientific wait. You could swap your labels and still have Newtonian Physics. And if you do you can have one where entropy increases (because we didn't change any dynamics) in time, but effects now reliably and consistently precede causes because we changed the labels.
You can now probably see the real reason they don't imply each other, they don't even use the same words. The second one never mentions causes, so it can't possibly tell us anything about causes. But for completeness, let's make an example where the first is true but the second is not.
Here we can label the causes and effects in the usual way (causes before effects). Now we need to look careful at what entropy is. We can imagine a nice classical system with nice time reversible dynamics (this doesn't always hold even for Newtonian Physics, but let's make a theory with very limited Force Laws where it does hold).
And the big thing is that entropy doesn't actually always have to increase even for an isolated system. For terminology let's be clear that a macrostate is an actual state of the system as it really is. And a macrostate is a set of microstates with the same macro description such as volume, pressure, and temperature. Microstates do not have entropy per se, macrostates do, and the entropy of a macrostate goes up when the the size of a macrostate goes up. And the size of a macrostate is nothing more than the number of microstates that are in the collection.
When a system evolves from a low entropy system to a high entropy system what happens is that the actual microstate initially belonged to a macrostate that was, say, large (i.e had many microstates in it) and evolved into a macrostate that was even larger (i.e. had even more microstates in it).
So clearly the actual microstate is one that can come from a macrostate with lower entropy. So you can setup a universe where say, the microstate is the opposite of the current microstate (e.g give every particle the opposite momentum), but all the dynamical laws run backwards (forces that used to be attractive are now repulsive, etc.) Now evolution in the forward time will make it evolve (in the future) to what the earlier time evolution would be in the original universe. So in this new universe we measure causes as before effects, but entropy decreases.
In reality, this is just like the first example where we relabelled the cause and effects, it's just that we also relabelled the direction time goes as well (that made all the momentums be opposite, and made attraction be repulsion and so forth).
So it's the same idea, except I'm explicitly pointing out that entropy can increases in the opposite direction as the time coordinate/parameter used in the laws of physics, as well as the direction that we label as cause and effect.
So what is the relationship?
Both are already implicitly using a preexisting time in their formulation, so they both have that in common. They are unrelated in the sense that it is possible for each to point in opposite directions.
There are other unaddressed issues. For instance to test theories you have to have controls and repetition. In both cases these interact in nontrivial ways with the principles. When you try to make sense of something you are trying to look at the information in terms of the things you can measure, and a small device operating fora limited duration can't know everything about everything. And our sense of control is seeped with particular assumptions that influence what we consider good enough. The sense of control influences what we colloquially call a cause, and affects what we consider spontaneous (which affects practical conceptions of entropy).
