Can you be a Christian without believing in the Resurrection of Christ?
No. "If Christ has not been raised, then our preaching is in vain and your faith is in vain" (1 Cor 15:14).
How did the disciples come to believe that Jesus is risen?
The disciples, who before had lost all hope, came to believe in Jesus' Resurrection because they saw him in a different way after his death, spoke with him, and experienced him as being alive.
The Easter events that took place in Jerusalem in the year 30 are not a made-up story. Following the death of Jesus and the defeat of their whole cause, the disciples fled ("We had hoped that he was the one to redeem Israel", Lk 24:21) or else barricaded themselves behind locked doors. Only their encounter with the risen Christ freed them from their paralysis and filled them with an enthusiastic faith in Jesus Christ, the Lord of life and death. (YOUCAT questions 104-105)
Dig Deeper: CCC section (631-638) and other references here.
Part 1: The Profession of Faith (26 - 1065)
Section 2: The Profession of the Christian Faith (185 - 1065)
Chapter 2: I Believe in Jesus Christ, the Only Son of God (422 - 682)
Article 5: "He descended into hell. On the third day He rose again" (631 - 658)
Jesus "descended into the lower parts of the earth. He who descended is he who also ascended far above all the heavens."476 The Apostles' Creed confesses in the same article Christ's descent into hell and his Resurrection from the dead on the third day, because in his Passover it was precisely out of the depths of death that he made life spring forth: Christ, that Morning Star, who came back from the dead, and shed his peaceful light on all mankind, your Son who lives and reigns for ever and ever. Amen.477
476.
477.
Roman Missal, Easter Vigil 18, Exsultet.
Paragraph 1: Christ Descended into Hell (632 - 637)
The frequent New Testament affirmations that Jesus was "raised from the dead" presuppose that the crucified one sojourned in the realm of the dead prior to his resurrection.478 This was the first meaning given in the apostolic preaching to Christ's descent into hell: that Jesus, like all men, experienced death and in his soul joined the others in the realm of the dead. But he descended there as Savior, proclaiming the Good News to the spirits imprisoned there.479
478.
Acts 3:15; Rom 8:11; 1 Cor 15:20; cf. Heb 13:20.
479.
Cf. 1 Pet 3:18-19.
Scripture calls the abode of the dead, to which the dead Christ went down, "hell" — Sheol in Hebrew or Hades in Greek — because those who are there are deprived of the vision of God.480 Such is the case for all the dead, whether evil or righteous, while they await the Redeemer: which does not mean that their lot is identical, as Jesus shows through the parable of the poor man Lazarus who was received into "Abraham's bosom":481 "It is precisely these holy souls, who awaited their Savior in Abraham's bosom, whom Christ the Lord delivered when he descended into hell."482 Jesus did not descend into hell to deliver the damned, nor to destroy the hell of damnation, but to free the just who had gone before him.483
480.
Cf. Phil 2:10; Acts 2:24; Rev 1:18; Eph 4:9; Pss 6:6; 88:11-13.
481.
Cf. Ps 89:49; 1 Sam 28:19; Ezek 32:17-32; Lk 16:22-26.
482.
Roman Catechism I, 6, 3.
483.
Cf. Council of Rome (745): DS 587; Benedict XII, Cum dudum (1341): DS 1011; Clement VI, Super quibusdam (1351): DS 1077; Council of Toledo IV (625): DS 485; Mt 27:52-53.
"The gospel was preached even to the dead."484 The descent into hell brings the Gospel message of salvation to complete fulfilment. This is the last phase of Jesus' messianic mission, a phase which is condensed in time but vast in its real significance: the spread of Christ's redemptive work to all men of all times and all places, for all who are saved have been made sharers in the redemption.
484.
Christ went down into the depths of death so that "the dead will hear the voice of the Son of God, and those who hear will live."485 Jesus, "the Author of life", by dying destroyed "him who has the power of death, that is, the devil, and [delivered] all those who through fear of death were subject to lifelong bondage."486 Henceforth the risen Christ holds "the keys of Death and Hades", so that "at the name of Jesus every knee should bow, in heaven and on earth and under the earth."487 Today a great silence reigns on earth, a great silence and a great stillness. A great silence because the King is asleep. The earth trembled and is still because God has fallen asleep in the flesh and he has raised up all who have slept ever since the world began... He has gone to search for Adam, our first father, as for a lost sheep. Greatly desiring to visit those who live in darkness and in the shadow of death, he has gone to free from sorrow Adam in his bonds and Eve, captive with him — He who is both their God and the son of Eve... "I am your God, who for your sake have become your son... I order you, O sleeper, to awake. I did not create you to be a prisoner in hell. Rise from the dead, for I am the life of the dead."488
485.
Jn 5:25; cf. Mt 12:40; Rom 10:7; Eph 4:9.
486.
Heb 2:14-15; cf. Acts 3:15.
487.
488.
Ancient Homily for Holy Saturday: PG 43, 440A, 452C; LH, Holy Saturday, OR.
IN BRIEF ⇡
By the expression "He descended into hell", the Apostles' Creed confesses that Jesus did really die and through his death for us conquered death and the devil "who has the power of death" (Heb 2:14).
In his human soul united to his divine person, the dead Christ went down to the realm of the dead. He opened heaven's gates for the just who had gone before him.
Paragraph 2: On the Third Day He Rose from the Dead (638 - 658)
"We bring you the good news that what God promised to the fathers, this day he has fulfilled to us their children by raising Jesus."489 The Resurrection of Jesus is the crowning truth of our faith in Christ, a faith believed and lived as the central truth by the first Christian community; handed on as fundamental by Tradition; established by the documents of the New Testament; and preached as an essential part of the Paschal mystery along with the cross: Christ is risen from the dead!
Dying, he conquered death;
To the dead, he has given life.490
489.
490.
Byzantine Liturgy, Troparion of Easter.