That is not correct. A closed system may by convention still exchange energy with the outside (just not any matter). What you're really talking about is called an isolated system, which is a theoretical construct that does not occur in nature.

Fundamentally, time, causality, and entropy are related, especially considering the idea of entropy as seen through a probabilistic lens with information causality in mind (see Pawlowski et al arXiv:0905.2292).

However, I think causality overall has a stronger, more basic link to the direction of time, and here is why: Entropy can always be reduced locally in an open system without affecting the flow of time at all. Causality, on the other hand, is not subject to reversal under any known circumstances, as long as a reasonable amount of information and time is involved. If, by some yet unknown mechanism, causality were reversed then so would be the direction of time, which in turn means the whole thing could not possibly be measured - not even in principle. Hence the argument that this reversal may not be possible since it's literally something that cannot exist in our universe. No such link exists between entropy and time, respectively.