[Certhas]

We need precision here. The intuitive notion of time that you start from has a lot of structure.

The question is where does this structure originate? Is there a fundamental thing "time" that has direction, irreversibility, a moment of now, memory of the past?

It doesn't appear so. Instead it seems that most of these properties emerge for complex systems.

If the universe was a box of gas there would be none of the above. So we conclude that time direction, memory, irreversibility, etc... are features of particular configurations of matter that we experience. A universe with the same laws of physics but without these features is conceivable, so they are not fundamental.