That's certainly what he meant and what the early disciples thought he meant.
1Co 5:6 Your glorying is not good. Know ye not that a little leaven leaveneth the whole lump?
1Co 5:7 Purge out therefore the old leaven, that ye may be a new lump, as ye are unleavened. For even Christ our passover is sacrificed for us:
1Co 5:8 Therefore let us keep the feast, not with old leaven, neither with the leaven of malice and wickedness; but with the unleavened bread of sincerity and truth.
A couple of points here. This was written to a predominantly gentile church. The context is clearly during passover and the days of leavened bread. The instruction is to celebrate, observe, the feast of passover and unleavened bread.
Later in the same letter:
1Co 11:23 For I have received of the Lord that which also I delivered unto you, That the Lord Jesus the same night in which he was betrayed took bread:
1Co 11:24 And when he had given thanks, he broke it, and said, Take, eat: this is my body, which is broken for you: this do in remembrance of me.
1Co 11:25 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.
Passover, the day that Jesus died, is an annual observance. Passover, the day that Jesus instituted the Lord's supper of wine and bread, was the days that his followers partook of the Lord's supper.
To believe that the Lord's supper was not intimately linked with Passover is to deny scripture, history and the culture of the time. The Lord's supper as practiced today has deviated greatly from how the Lord and his original disciples practiced it.
Paul is exhorting his listeners to partake of Holy Eucharist. He is, after all, an early father of the Catholic Church.
Yes and no.
The unleavened bread that was broken and eaten was after the first 2 glasses of wine had been offered and the passover meal had been eaten. It was the 3rd glass of wine that was held up by Jesus and offered and drunk in remembrance of Him.
The point being that even if the prior 2 glasses of wine are not offered and the meal is not eaten (what became the Agape Feast) partaking of the unleavened bread and drinking from the 3rd glass is the key point in the LORD's SUPPER.