Well, unless you are using a poor translation (such as that from the Watchtower Society - a translation no professional Greek or New Testament scholar outside of that society would agree is very accurate) there is a very strong argument that the New Testament documents do claim Jesus is divine, aka, God. The usage of the term "kurios" throughout, some of Paul's letters and the first chapter of John (among other reasons) are used to show that Jesus is divine and the earliest Christian disciples believed it to be so. For those interested in more, see the Doctrine of Trinity podcasts here on RF.org as well as Dr. Craig's chapters on this in his books Reasonable Faith and Philosophical Foundations for a Christian Worldview. Norman Geisler and N.T. Wright have material that also give arguments for Jesus' divinity as well (obviously, this is just a small sampling of contemporary material that argue for Jesus' divinity - you can even read the ante-Nicene writings of the early church fathers for material as well).
But, with that said, let's not derail the question posed in the OP. Let's assume Jesus is divine and thus omniscient for this exercise (the question of Jesus' divinity can be left as another exercise elsewhere).
> What is the best responsto when someone who doubts Jesus'
> omniscience brings up Matthew 24:36 when He said not even He knew the
> day or hour?
First, that can go towards arguing that Jesus was truly human - which is something all orthodox Christians also believe! We believe Jesus was truly human and truly divine - this just goes to prove the human part

Now, I personally think Dr. Craig presents a very rational, plausible and possible model for Jesus's divinity (and specifically his omniscience) in his incarnate state. Rather than me regurgitate his full thoughts on how Jesus can be divine yet not know the time of his return, I recommend the following material.You will want to listen to the podcasts for
Defenders 2, Doctrine of Christ, parts 1 through 8. You can also read very similar material in Philosophical Foundations for a Christian Worldview, Chapter 30, "The Incarnation". If you want to go directly to the part that answers your question, listen to
Defenders 2 podcast, Doctrine of Christ part 7 where he directly answers this question that he puts forth:
"This still leaves unexplained the problem of providing a realistic understanding of Jesus as we read about him in the Gospels. If Jesus was the Logos in the flesh, then why wasn’t he omniscient? Why didn’t he know everything? How could he be tempted? If he’s the second person of the Trinity he would just blow these temptations off like smoke. How do we understand the human consciousness of Jesus?"
In short, I will provide a quick summary of the three planks of Dr. Craig's possible model for the incarnation that can answer your question. Listen to the podcasts or read his material for in depth discussions:
1) We postulate (with the Council of Chalcedon) that in Christ there is one person who exemplifies two distinct and complete natures, one human and one divine.
2) We postulate (with the early church father Apollinarius) that the Logos (the Word, the second person of the Trinity) was the rational soul of Jesus of Nazareth
...and the important part that answers your question...
3) We postulate that the divine aspects of Jesus' personality were largely subliminal during his state of humiliation/incarnation.
Again, I highly recommend that you read Dr. Craig's materials or listen to his podcasts to get more in-depth information on these three planks, what they mean and why he proposes them.