How does the Passover foreshadow the coming of Jesus as the Messiah? In 1 Corinthians 5:7, Paul refers to Jesus as "Christ, our Passover", suggesting that the Passover of Exodus somehow foreshadowed the life of Jesus.  

How specifically does the Passover do this?


  Cleanse out the old leaven that you may be a new lump, as you really
  are unleavened. For Christ, our Passover lamb, has been sacrificed.  1
  Corinthians 5:7 ESV
In regards to Passover, Jesus is closely identified with the Passover lamb.

Exodus 12 gives the specific details on how to select the lamb and how it was to be used.


  3 Announce to the whole community of Israel that on the tenth day of
  this month each family must choose a lamb or a young goat for a
  sacrifice, one animal for each household. ... 5 The animal you select
  must be a one-year-old male, either a sheep or a goat, with no
  defects. 6 Take special care of this chosen animal until the evening of the fourteenth day of this first month. Then the whole assembly of the community of Israel must slaughter their lamb or young goat at twilight. 7 They are to take some of the blood and smear it on the sides and top of the doorframes of the houses where they eat the animal. ... 13 But the blood on your doorposts will serve as a sign, marking the houses where you are staying. When I see the blood, I will pass over you. This plague of death will not touch you when I strike the land of Egypt.


A couple of key points here,


The lamb was to be without defects
The blood covered the doorposts and it was because of this blood that those within the home were saved from death.


Jesus as we know was/is perfect (without sin) in every way and it's because of his shed blood that we are saved as well.

In John 1:29 John the baptist makes this declaration about Jesus:


  The next day John saw Jesus coming toward him and said, “Look! The
  Lamb of God who takes away the sin of the world!


And in 1 Peter 1:19-20


  It was the precious blood of Christ, the sinless, spotless Lamb of
  God. 20 God chose him as your ransom long before the world began, but
  he has now revealed him to you in these last days


.