Were the Jews expecting the Messiah to suffer at all? When Jesus spoke about his future suffering it did not seem to register properly.

For example:


  (Matthew 16:22)
  Peter took him aside and began to rebuke him. “Never, Lord!” he said. “This shall never happen to you!”


This raises the question: Did the Jews push the suffering of the servant Messiah (Isaiah 53) to the background of their mind, or was there no expectation that Messiah would suffer at all?
It does seem that the Jews hadn't understood that the Messiah would have to suffer, or that the suffering servant of Isaiah 53 was the Messiah. Apart from the verse you have quoted, there are other verses which suggest that the Jews didn't understand this. 

They understood that the Christ would be the "King of Israel" (Matthew 15:32), they knew that he would be the one to govern the people of Israel (Matthew 2:6), but they didn't understand that his kingship was not of this world (John 18:36). They expected a political Messiah, one who would save them from their enemies in this world, not one who would "save his people from their sins" (Matthew 1:21). This is why, King Herod was troubled and all Jerusalem with him (Matthew 2:3), for he thought that the Christ would come and take his political throne from him. 

Even after having the knowledge that he was indeed the Christ (Matthew 16:16), Peter rebuked him for saying that he would have to suffer. Even after he actually suffered and died, two of his disciples seemed to have no clue what was happening and Jesus said to them, 

"O foolish men, and slow of heart to believe all that the prophets have spoken! Was it not necessary that the Christ should suffer these things and enter into his glory?" (Luke 24:25-26)

And then Jesus had to interpret the scriptures for them, to show them all the things in the scriptures about the Christ (Luke 24:27). 

So it does seem that to the Jews, it wasn't plain and clear from the scriptures that the Messiah had to suffer and in fact the evidence shows that they were expecting him to rule like a political king and not undergo suffering.