Yes, you are right, the name "Son of Man" does refer to Jesus, and I am glad you have thought about the little incident I recorded. It was one, which the disciples did not talk about much, yet it had a profound effect on them. At first they did not understand it as I record in my gospel. Not that they were surprised that someone would try to kill Him, but they thought a man of His abilities could easily avoid being killed - He had done it before (Luke 4: 28-30). It was only after the resurrection that Jesus explained things to them. (Luke 24: 44-49).
There are two things about Jesus you must remember. Firstly He truly is the Son of God and He knows the purposes and plans of His Father. Because of this He spoke with absolute authority, which infuriated the Pharisees and thrilled His followers -until He began to talk about His death.
The disciples should have believed all His words, but like so many of us, they only believed what suited them. It was not until after Pentecost that they were able to understand and believe all that He had told them. Of course they had to believe the prophesy of His death because it happened - as did the resurrection.
The second thing about Jesus was His attitude to the Jewish scriptures. He saw in these God speaking to people about Himself, sometimes directly sometimes in parables, and He often used the scriptures to confirm what He was saying. In this way He related the Jewish rituals of sacrifice for the atonement of sin to His death as the Passover lamb for the sins of the whole world; and He saw himself as the suffering servant of the prophesy of Isaiah, a set of prophesies which the Pharisees despised because they wanted a military king.
Chapter 53 of Isaiah is good to read because until Jesus died it seemed so wrong, now it seems so right. Many are the prophesies which He fulfilled, some deliberately, like riding into Jerusalem on a donkey (Zechariah 9: 9) and some involuntarily such as His burial in a rich man's tomb (Isaiah 53: 9) - and we do well to study these ancient scriptures because many of the prophesies have yet to be fulfilled.
I think either of these would be adequate to explain Jesus' knowledge of His death, but it is a principle of Jewish law that a verdict must be based on the evidence of at least two witnesses, a principle Jesus was always careful to observe. In this case the "witnesses" were the scriptures and His own prophetic revelation from God.