Copy from your script
John 1:51
"And he said to him, 'Truly, truly, I say to you, you will see heaven
opened, and the angels of God ascending and descending on the Son of
Man'". ESV.
Genesis 28:12
"And he dreamed, and behold, there was a ladder set up on the earth,
and the top of it reached to heaven. And behold, the angels of God
were ascending and descending on it!
In comparison of these two verses, Jesus substituted "the Son of Man" to "it - i.e. the ladder between the earth and heaven". Jesus was actually claiming himself is the bridge between the human and the Lord.
Worth to note that at that time, Jesus told Nathanael this vision when Nathanael confessed Jesus is the Son of God, the king of Israel. So what is the connection between the account of Nathanael and the dream of Jacob?
The life of Jacob had many accounts of deceit.
- Jacob deceived his father Isaac to earn his blessing (Gen chapter 27)
- Laban deceived Jacob his marriage, in order to prolong his service to him. (Gen chapter 29-30)
- Jacob deceived Laban by not telling him he was running away (Gen 31:20)
- Jacob deceived his brother Esau to go with him but instead he went to Sukkoth. (Gen chapter 33)
- Jacob's son deceived Shechem and Hamor to get all males circumcised. While they were in pain, they got killed by Jacob's son Simeon and Levi. (Gen chapter 34)
- Jacob's son deceived Jacob that Joseph was killed by ferocious animal (Gen 37:33)
But how did Jesus called Nathanael on their first met? John 1:47 read;
47 Jesus saw Nathanael coming toward him and said of him, “Behold, an Israelite indeed, in whom there is no deceit!” (ESV)
Jacob was not born to deceive but many kind of deceit followed his life, yet he was blessed by the Lord. In his dream he had a vision of angel ascending and descending on a ladder. So when Jesus told Nathanael that he was a true Israelite of no deceit, that Jesus knew him already when he was under the fig tree. Nathanael immediately confessed as only true God knew his heart. Jesus further reiterated the dream of Jacob as his dream is time to fulfill, for Jesus is the ladder, the mediator, the advocate, the bridge between the human and the Lord. And this is "On the Son of Man" meant in John 1:51.
Ascending and Descending
Personally I don't see the order of these has a specific meaning, though I generally accept all kinds of spiritual comments to it.
We should note our relation with the Lord is never just one direction. The interaction is a cycle, that makes no difference which direction to go first. So why "Ascending" preceded "Descending"? Perhaps Book of Job had a clue.
Job 1:6-7 (ESV)
6 Now there was a day when the sons of God came to present themselves
before the Lord, and Satan also came among them.
7 The Lord said to Satan, “From where have you come?” Satan answered the Lord and said, “From going to and fro on the earth, and from walking up and down on it.”
Some English translation will use "going back and forth on it" e.g. NIV. But they all indicate that the angels of the Lord are on earth, then come up to present themselves before the Lord, and go back down to the earth.