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That is not supported in scripture. There are a number of passages that discuss salvation and none of them involve bishops or the catholic church...they involve faith, belief and baptism.

Authority of the apostles:

Then he summoned his twelve disciples and gave them authority over unclean spirits to drive them out and to cure every disease and every illness.… Whoever will not receive you or listen to your words--go outside that house or town and shake the dust from your feet.… Whoever receives you receives me, and whoever receives me receives the one who sent me.
Matt. 10:1,14,40
Whatever town you enter and they do not receive you, go out into the streets and say, 'The dust of your town that clings to our feet, even that we shake off against you.' Yet know this: the kingdom of God is at hand. I tell you, it will be more tolerable for Sodom on that day than for that town. "Woe to you, Chorazin! Woe to you, Bethsaida! For if the mighty deeds done in your midst had been done in Tyre and Sidon, they would long ago have repented, sitting in sackcloth and ashes. But it will be more tolerable for Tyre and Sidon at the judgment than for you. And as for you, Capernaum, 'Will you be exalted to heaven? You will go down to the netherworld.'" Whoever listens to you listens to me. Whoever rejects you rejects me. And whoever rejects me rejects the one who sent me."
Luke 10:10-16
Amen, amen, I say to you, whoever receives the one I send receives me, and whoever receives me receives the one who sent me.
John 13:20
(Jesus) said to them again, "Peace be with you. As the Father has sent me, so I send you." And when he had said this, he breathed on them and said to them, "Receive the holy Spirit. Whose sins you forgive are forgiven them, and whose sins you retain are retained."
John 20:21-23

The office of apostles continues after the death of the original twelve:

During those days Peter stood up in the midst of the brothers (there was a group of about one hundred and twenty persons in the one place). He said, "My brothers, the scripture had to be fulfilled which the holy Spirit spoke beforehand through the mouth of David, concerning Judas, who was the guide for those who arrested Jesus. He was numbered among us and was allotted a share in this ministry. He bought a parcel of land with the wages of his iniquity, and falling headlong, he burst open in the middle, and all his insides spilled out. This became known to everyone who lived in Jerusalem, so that the parcel of land was called in their language 'Akeldama,' that is, Field of Blood. For it is written in the Book of Psalms: 'Let his encampment become desolate, and may no one dwell in it.' And: 'May another take his office.' Therefore, it is necessary that one of the men who accompanied us the whole time the Lord Jesus came and went among us, beginning from the baptism of John until the day on which he was taken up from us, become with us a witness to his resurrection." So they proposed two, Joseph called Barsabbas, who was also known as Justus, and Matthias. Then they prayed, "You, Lord, who know the hearts of all, show which one of these two you have chosen to take the place in this apostolic ministry from which Judas turned away to go to his own place." Then they gave lots to them, and the lot fell upon Matthias, and he was counted with the eleven apostles.
Acts 1:15-25

Catholic teaching on the Eucharist:

(Jesus said) "I am the living bread that came down from heaven; whoever eats this bread will live forever; and the bread that I will give is my flesh for the life of the world." The Jews quarreled among themselves, saying, "How can this man give us (his) flesh to eat?" Jesus said to them, "Amen, amen, I say to you, unless you eat the flesh of the Son of Man and drink his blood, you do not have life within you. Whoever eats my flesh and drinks my blood has eternal life, and I will raise him on the last day. For my flesh is true food, and my blood is true drink. Whoever eats my flesh and drinks my blood remains in me and I in him. Just as the living Father sent me and I have life because of the Father, so also the one who feeds on me will have life because of me. This is the bread that came down from heaven. Unlike your ancestors who ate and still died, whoever eats this bread will live forever."
John 6:51-58
While they were eating, Jesus took bread, said the blessing, broke it, and giving it to his disciples said, "Take and eat; this is my body." Then he took a cup, gave thanks, and gave it to them, saying, "Drink from it, all of you, for this is my blood of the covenant, which will be shed on behalf of many for the forgiveness of sins.
Matt. 26:26-28