No, the universe does not have an absolute time. neither does it have an absolute space. What it has is spacetime: a combination of the two into a single geometry. That is the whole point of relativity.
In a sense, again using high school algebra, asking if there is an absolute time is like asking if a plane has an 'absolute y axis'. And the answer is no.
The term 'passage of time' is a subjective one. At least in relativity ALL of time and ALL of space exists as a single geometry. So past, present, and future all are part of this geometry. The subjective feel of the passage of time is linked to the 'arrow of time' and the changes in entropy between different points on a world line (a path through spacetime).
Spacetime is a mathematical model, a concept, it represents reality but it is not reality, That is where many scientists go wrong, they get so involved in mental models that they forget that the models are only a mental representation of something actually real. It is as though Michael Angelo began to think his sculptures were equal to that which was sculptured.
Now because there is no such thing as absolute now in the mathematical models, you imagine that reality itself, existence itself, has not an absolute now. How can existence itself, change its rate of its continuing to exist. There is no time per say, man observes some regular aspect of reality, planets stars, pendulums, oscillations, etc. as a proxy measurement of the passage of the continuation of existence, whatever the form.
Now I am not saying that our senses don't create the impression of 'time', I say the creation of a measurement of rate of change around us using stars and pendulums were great inventions. But beyond this, there is only the absolute now always being now, that creates the impression of a flow from past to future, only because of an observer..
The pendulum swings 60 times at the rate of one second per swing, it means the universal now was present and continued to be present for 60 swings of the pendulum.