Some people do assume that, when Jesus spoke of the "kingdom", he was referring to his return on the Judgment Day. I used to assume that myself. I was wrong. Eventually someone showed me that, contrary to my assumptions about the "kingdom", the Christians of the first century believed that the "kingdom" had already come in their day. For example, consider Colossians 1:13,14, "He has delivered us from the domain of darkness and transferred us to the kingdom of his beloved Son, in whom we have redemption, the forgiveness of sins." The first century Christians were described as having ALREADY been transferred to the "kingdom".
Similarly, Revelation 1:5 says that Jesus has "made us a kingdom, priests to his God and Father" and, in verse 9 of that chapter, the writer describes himself as "John, your brother and partner in the tribulation and the kingdom". Since the Christians of the first century described the "kingdom" was something that was already present then, and said that they were already in the "kingdom", I had to admit that my assumption about the meaning of "kingdom" was wrong.
Politically speaking, those who are under the authority of a king are his kingdom. Those who are in rebel-held territory are not in his kingdom (though perhaps they should be). It is no different in the New Testament. Those who are under the authority of the King are in His Kingdom. This world, as you have no doubt noticed, is rebel-held territory. Here and there in this "break-away republic" we call Earth, there were (and are) loyalists who are citizens of the Kingdom.
In the New Testament, contrary to what I assumed, Christ's return in judgment is not the beginning of his kingdom. It is just the opposite. It is the END of his kingdom. I Corinthians 15:24-26 says, "Then comes the end, when he delivers the kingdom to God the Father after destroying every rule and every authority and power. For he must reign until he has put all his enemies under his feet. The last enemy to be destroyed is death."
With this as background, it is easy to understand Matthew 16:28, "Truly, I say to you, there are some standing here who will not taste death until they see the Son of Man coming in his kingdom."
What Jesus said that day was not hyperbole or mysticism at all. It was actually an understatement. Of those to whom Jesus was speaking, most (apparently all but Judas) lived to "see the Son of Man coming in his kingdom" and then lived their lives as citizens loyal to the King who was "not of this world" (John 18:36).