Looking at Mark 9:1 and 1 Cor. 15:51, it is clear that both Jesus and Paul were mistaken in expecting,  the arrival of the Kingdom of God in the case of Jesus   and in Paul’s case, the second coming, in the lives of some of those present in each situation. How do you cope with that ?

I was just thinking of you this week.  I hope you are well.  I will be in Europe this summer to teach, but only in Hungary, Romania, Bulgaria, Sweden and Norway–not London.
This is not hard to "cope" with at all because in both cases you misunderstand the speaker. 
In Mark 9:1 Jesus is talking about the promised coming of the kingdom of God which happened in Acts 2:1-41 on the Day of Pentecost when the Holy Spirit came with power.  This "coming" of the kingdom was prophesied in Joel 2:28-32, Isaiah 2:2-4 Daniel 2:44 and many other Old Testament passages.  John the Baptist also predicted the coming of the Kingdom (Matthew 3:1) at Pentecost.  In Acts 1 Jesus was talking after his resurrection about the kingdom, telling his disciples to wait in Jerusalem to wait for the coming of the kingdom "with power."  All this was fulfilled when the Spirit was poured out on the apostles, wind, "fire" and speaking in tongues were signs of a new aspect of the Kingdom of God as anticipated throughout the Old Testament and as prophesied by Jesus in Mark 9:1.  It is interesting to note that Jesus said some of the disciples would still be alive.  In fact, all of them were except Judas, who had killed himself before the prophecy was fulfilled.  If you look at Mark 9:1, Jesus definitely is not saying that he was going to come back (as prophesied in Matthew 24:30-35) while some of the disciples were still alive.
In 1 Corinthians 15:51 Paul is talking about something completely different, as you notice.  To be honest, although I can easily see how one not really familiar with the New Testament can make the mistake of seeing an error in Mark 9:1, I am surprised you use this example as if Paul is making a mistake.  Paul says, "Listen, I tell you a mystery: We will not all sleep, but we will all be changed."  Here, Paul is talking about Christians in general.  Not all Christians will die, but all will be changed.  This is still true.  According to Paul (1 Thess 4) and Jesus (Luke 22) when Jesus comes back, those who are still alive in Christ will be taken to be with him, and those who have died in Christ will be raised.  I really do not see how you can see a mistake on Paul’s part in 1 Cor 15:51.  I suppose that if he had said, Listen, I tell you a mystery: some of us who are still alive today will not fall asleep…. that would be problematic.  How else do you propose Paul should have said this?  It is true of Christians, that "we will not all sleep."
I will grant you that in other passages besides 1 Cor 15 there is some evidence that Paul may have believed that Jesus would come back during the lifetime of some of the disciples living at that time, but he never says this in any of his letters.  Of course, the Bible is inspired by God, so obviously Paul’s opinion never entered the text, assuming that was his opinion, which we cannot be sure about. 
Jesus said about the time of his return (Matthew 24:36) that no one knows the day or the hour of his return.  This applies to Paul as well.  Paul taught all Christians that they should be ready for the coming of Jesus.  Peter said that God is not slow in keeping his promise–that to him a day is like a thousand years and a thousand years is like a day.  When will Jesus come back?  I do not know, but your attempt to find an error in the New Testament is a failure in this case.
John Oakes

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