How do we argue when someone posits a non-causal universe So, I've been thinking about apologetics recently, and I've come to a rather nasty knot which I've not been able to unwind. It is relatively easy to get from the idea, "there must have been a first thing" to God if we assume causality, but what if someone is not willing to stipulate causality? How do we argue for God when we cannot even agree on causality?
Do you mean that this person rejects the idea of causality itself? i.e. that he believes that things happen with no cause? Or that he rejects the idea that there must be a First Cause?

If he rejects causality, he is rejecting science as well as Christianity. You could, perhaps, propose experiments to show that the universe is predictable. Like, I hold a rock at arms length and let it go. Every time I perform this experiment it falls to the ground: it never hovers motionless in midair or flies off at an angle. Etc.

If he postulates an infinitely old universe as an alternative to a First Cause, I'd point out the Second Law of Thermodynamics: entropy. Energy can neither be created nor destroyed, but it tends to become more evenly distributed over time. After some amount of time, all the energy in the universe will become completely evenly distributed. As doing useful work requires energy differentials, at this point the universe would be dead. Physicists refer to this as "heat death". We can debate how long this will take, but it is a finite amount of time, so if the universe was infinitely old, it would have reached this state.

Some people try to get around this by postulating speculative theories, but none of these have any foundation in science: they are pure philosophical speculation. As far as the evidence goes, entropy is a well-established theory.