The Passover itself is a type and shadow of things to come. The unblemished paschal lamb typifies Jesus Christ. The blood of the lamb, serving to deliver Israel from bondage, points to the atoning blood that delivers all mankind from spiritual bondage.
The most exact way to fulfill the foreshadowing of Passover would be for the crucifixion to happen at the same time the Passover lambs were being killed - on the 14th toward the evening near the temple. John's account of the last supper holds true to this timeline (John 13:1, 18:28) as he specifically mentions the last supper as happening before the feast of the Passover and the officers of the Jews not wanting to accompany Jesus to the tribunal before Pilate due to their not wanting to be defiled before Passover.
The Contradictions
Luke says, "then came the day of unleavened bread, when the Passover must be killed." Mark says, "And the first day of unleavened bread, when they killed the Passover, his disciples said unto him, Where wilt thou that we go and prepare that thou mayest eat the Passover?" (Mark 14:12). Matthew reads almost identically (Matthew 26:17). Why do Matthew and Mark refer to the last supper as a Passover meal while John refers to it as a day before Passover? We can understand all these apparent contradictions in three ways:
First, the actual first day of the Feast of Unleavened Bread became synonymous with both the day of preparation for the feast and also the time of year for the feast. Think of this similar to how we might say we had Christmas dinner with the family which may have been on Christmas Day, Christmas Eve, or sometime in the Christmas season.
Secondly, this can more accurately be translated, "Now before [the Feast of] Unleavened Bread the disciples came to Jesus..." or "And for the sake of the first day of the festival of Unleavened Bread the disciples came to Jesus..." The question seems to be asked in regards to the preparation day rather than on that day.
Thirdly, given that the last supper started on after sundown, it technically started on the 14th of Nisan. The Passover lambs were slain on the 14th before sundown (between noon and 3pm). Therefore, the Last Supper was actually eaten on the day "when they killed the Passover"
The disciples ask about the Passover preparation on the afternoon of the 13th (this would be some time during the day on Tuesday) and Jesus tells them a place in the city is prepared. After sundown, which now makes it the 14th, Jesus came to them in the evening (Mark 14:17). Their shared meal does actually happen on the same day the Passover is killed (the 14th) and the day before the Passover seder and feast of Unleavened Bread (the 15th which starts at sundown).
This is the meal that Paul writes of - on the same night he was betrayed. "That the Lord Jesus the same night in which he was betrayed took bread: And when he had given thanks, he brake it, and said, Take, eat: this is my body, which is broken for you: this do in remembrance of me. After the same manner also he took the cup, when he had supped, saying, This cup is the new testament in my blood: this do ye, as oft as ye drink it, in remembrance of me." (1 Corinthians 11:23-25).
By the way, when Paul uses the word "bread", he uses the Greek for regular bread (artos), not for unleavened (azyma). If the Last Supper was actually a Passover meal, unleavened bread would have been used.
As the Last Supper began, Jesus told the disciples that he desired to eat the Passover with them, but will not be able to until it is fulfilled in the Kingdom of God (Luke 22:15-16). In other words, he hints to them that even though they have spoken of preparation for the Passover, he will not be there.
The Passover was a family-oriented commemoration. The priests had already discussed arresting Jesus before the feast, as there would be a riot if they acted during the feast. John confirmed that the crucifixion was on the "day of Preparation for the Passover".
If the Last Supper was an actual Passover seder, then the chief priests would not have interrupted their own family traditions to seek Jesus and His trial and crucifixion would have to take place during the week long Feast of Unleavened Bread; both of these actions are highly unlikely as they would have been forbidden by Jewish law.
In summary, the Last Supper was not a Passover meal, but happened 24 hours before the official Passover meal.
The Fast of the Firstborns
In Judaism there is a special meal on the day before Passover called "The Fast of the Firstborns". This tradition was held by both the Galileans and the Samaritans. As Galileans, it would have been natural for Jesus and his disciples to celebrate the Fast of the Firstborns, or in Hebrew, seudah maphsehket, which literally translates to "Last Supper".
This celebration remembers the impact on the firstborns in Egypt, both those who were saved and protected and those Egyptians who were killed and calls for a day of fasting for firstborn sons to honor the Lord. The beginning of this day of fasting would be the first meal of the day, that is, the nighttime meal.
It is fitting that the Fast of the Firstborns would be where The Firstborn Son would establish a new covenant and begin his fast. "And he took the cup, and gave thanks, and gave it to them, saying, Drink ye all of it; For this is my blood of the new testament, which is shed for many for the remission of sins. But I say unto you, I will not drink henceforth of this fruit of the vine, until that day when I drink it new with you in my Father's kingdom." The next time He would commune with them would be after the fulfillment of his mission.