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?
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The question (currently) is unclear, but I think I can give some general advice for you to try out:
Personally, I've never found this type of apologetic to be helpful in "converting" someone who does not share my fundamental beliefs. We need Christian philosophers to engage on these topics as Christians, but ultimately it's God who will be able to convince people He exists, not us.
This is a logical impossibility. If time had no beginning and no Beginner, then there is an infinite amount of time before now. If there is an infinite amount of time before now, we would never get to today, since you cannot traverse an infinite amount of anything.
So, it's logically absurd to suggest that time had no beginning. It did have a beginning and therefore has a cause (that is outside of time--and space and matter).
If someone is embracing logical absurdities, then rational discussion is probably not possible. You may as well suggest there both was and was not a first thing.
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.
Those that are not impressed with God's early work may be impressed with His later doings.
If find such a person who can get through Mark's Gospel without appreciating the earnestness with which the events in Jesus mission are portrayed; the rapidity of His miracles and signs; the wisdom which He chose to codify as His Gospel of love, then you've got a hard case and can probably wipe the dust from your sandals and move on.
I'm just saying, as much as its a good thing to believe in the infiteness of God, simply accepting that and becoming a deist is not the most grace ridden method for salvation.