Death was introduced after the original sin (Adam was immortal until the fall), some people even think that Our Lady didn't die but just was assumpted in her sleep.
If Christ didn't have the original sin, then how did He die?
Jesus voluntarily died so that the rest of humanity can inherit his righteousness and thus eternal life:
God made him who had no sin to be sin for us, so that in him we might become the righteousness of God. (2 Cor. 5:21)
though he existed in the form of God, did not regard equality with God as something to be grasped, 7 but emptied himself, taking the form of a slave, assuming human likeness. And being found in appearance as a human, 8 he humbled himself and became obedient to the point of death— even death on a cross. (Philippians 2)
Many assume that "Adam was immortal until the fall", but had that been the case, then there would have been no need to debar him from the way to the Tree of Life. Had he partaken of the Tree of Life, then he could never have died. Therefore, God prevented him getting to it after he had disobediently partaken of the other tree, the one that gave death. This is worth mentioning at the start of this answer, because if a wrong understanding of Adam is entertained, that will hinder a correct understanding of "the last Adam", Jesus Christ,.
The New Testament explains that the first Adam was made a living soul, and was of the earth (not spiritual) but the last Adam is from heaven and is a life-giving (quickening, K.J.V.) spirit - 1 Corinthians 15:45-47. We are also told that Adam was the figure of him who was to come from heaven - the uncreated, eternal Son of God (Romans 5:14). This means that a physical body was "prepared" for the Son of God by God, in fulfillment of prophecies (Hebrews 10:5) and it was this human body that could - and did - die physically. Christ did not just appear to have a human body. He was fully human. All physical, human bodies can die. Even if Jesus had suffered a fatal accident during his life, his body would have died. But God had given his angels command over him that that should not happen. It was even foretold that not a bone of his body would be broken. Several times, when people tried to stone Jesus to death, he slipped away from them, "for my time has not come" he said.
The appointed time for him to die came, and he knew when that was, and gave himself over to wicked men who, he knew, would crucify him (see Matthew 26:2). That was the foreordained way for the sinless Son of God to become God's sacrificial 'Lamb'. He died physically because physical means were used to take his physical life. Jesus could have resisted; he could have called upon his Father to send more than 12 legions of angels to prevent him being crucified - Matthew 26:53-54. But he did not resist. Instead, he submitted to the will of God, that he die in the flesh as the only perfect sacrifice for sin there has ever been.
He could die, because he did not merely appear to have a human body - he was fully human. Yet due to also being fully God, he entered the world without sin, resisted all temptation, and remained sinless till he knew the time had come to give up his spirit to God in death - Luke 23:46. Jesus was not a martyr; his death was not an unfortunate accident. He deliberately gave himself over to death at the exact time ordained. But because death can only claim sinners, the grave could not retain him, and he arose in triumph out of the grave - Acts 2:23-27.
If Jesus Christ was born without the original sin, then how could He have died?
Jesus was totally human in all things except sin. Pain, suffering and death are part of life! Being free of original sin would preserve Christ from illness and disease, but not pain, suffering and/or death.
Plants and animals do not have original sin, yet they too are subject to death and decay.
Original sin should not be seen to mean that one without any stain of original sin one could not die physically; but rather that one would die in a more spiritual sense. That is to say death of the soul in the sense of being deprived of sanctifying grace and meaning that without baptism, the soul can not be saved: thus a spiritual death! One can not have life in God and share in the Beatific Vision. Jesus never lost this condition.
Jesus never suffered any of the conditions of original sin and had a perfect physical conditioned body. He could never become sick or die to illness, but could be put to death by the ungodly and those living in the conditions of original sin. After all was his body not consistent with flesh and blood, like all men?
Christ being born without original sin and had never been guilty of actual sin was never deprived of the Beatific Vision which as the Second Person of Most Holy Trinity. The idea of original sin does not necessarily mean to preserve one from a uniquely physical death!
Death and suffering although may be seen as physical evils, they are not to be seen as sin. Death and original sin must be understood as two distinct things that entered the world because of Adam.
Jesus’s human soul remain completely and totally united to God. His soul in it’s human was never deprived of sanctifying grace and as such he never could experience ”the death of the soul”!
Nature of original sin
This is a difficult point and many systems have been invented to explain it: it will suffice to give the theological explanation now commonly received. Original sin is the privation of sanctifying grace in consequence of the sin of Adam. This solution, which is that of St. Thomas, goes back to St. Anselm and even to the traditions of the early Church, as we see by the declaration of the Second Council of Orange (A.D. 529): one man has transmitted to the whole human race not only the death of the body, which is the punishment of sin, but even sin itself, which is the death of the soul [Denz., n. 175 (145)]. As death is the privation of the principle of life, the death of the soul is the privation of sanctifying grace which according to all theologians is the principle of supernatural life. Therefore, if original sin is "the death of the soul", it is the privation of sanctifying grace.
The Council of Trent, although it did not make this solution obligatory by a definition, regarded it with favour and authorized its use (cf. Pallavicini, "Istoria del Concilio di Trento", vii-ix). Original sin is described not only as the death of the soul (Sess. V, can. ii), but as a "privation of justice that each child contracts at its conception" (Sess. VI, cap. iii). But the Council calls "justice" what we call sanctifying grace (Sess. VI), and as each child should have had personally his own justice so now after the fall he suffers his own privation of justice.
We may add an argument based on the principle of St. Augustine already cited, "the deliberate sin of the first man is the cause of original sin". This principle is developed by St. Anselm: "the sin of Adam was one thing but the sin of children at their birth is quite another, the former was the cause, the latter is the effect" (De conceptu virginali, xxvi). In a child original sin is distinct from the fault of Adam, it is one of its effects. But which of these effects is it? We shall examine the several effects of Adam's fault and reject those which cannot be original sin:
(1) Death and Suffering.- These are purely physical evils and cannot be called sin. Moreover St. Paul, and after him the councils, regarded death and original sin as two distinct things transmitted by Adam.
(2) Concupiscence.- This rebellion of the lower appetite transmitted to us by Adam is an occasion of sin and in that sense comes nearer to moral evil. However, the occasion of a fault is not necessarily a fault, and whilst original sin is effaced by baptism concupiscence still remains in the person baptized; therefore original sin and concupiscence cannot be one and the same thing, as was held by the early Protestants (see Council of Trent, Sess. V, can. v).
(3) The absence of sanctifying grace in the new-born child is also an effect of the first sin, for Adam, having received holiness and justice from God, lost it not only for himself but also for us (loc. cit., can. ii). If he has lost it for us we were to have received it from him at our birth with the other prerogatives of our race. Therefore the absence of sanctifying grace in a child is a real privation, it is the want of something that should have been in him according to the Divine plan. If this favour is not merely something physical but is something in the moral order, if it is holiness, its privation may be called a sin. But sanctifying grace is holiness and is so called by the Council of Trent, because holiness consists in union with God, and grace unites us intimately with God. Moral goodness consists in this, that our action is according to the moral law, but grace is a deification, as the Fathers say, a perfect conformity with God who is the first rule of all morality. (See GRACE.) Sanctifying grace therefore enters into the moral order, not as an act that passes but as a permanent tendency which exists even when the subject who possesses it does not act; it is a turning towards God, conversio ad Deum. Consequently the privation of this grace, even without any other act, would be a stain, a moral deformity, a turning away from God, aversio a Deo, and this character is not found in any other effect of the fault of Adam. This privation, therefore, is the hereditary stain. - Original Sin (Catholic Encyclopaedia)
Let us ponder the words of St. Paul in Romans 5:12-21:
12 Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned:
13 (For until the law sin was in the world: but sin is not imputed when there is no law.
14 Nevertheless death reigned from Adam to Moses, even over them that had not sinned after the similitude of Adam's transgression, who is the figure of him that was to come.
15 But not as the offence, so also is the free gift. For if through the offence of one many be dead, much more the grace of God, and the gift by grace, which is by one man, Jesus Christ, hath abounded unto many.
16 And not as it was by one that sinned, so is the gift: for the judgment was by one to condemnation, but the free gift is of many offences unto justification.
17 For if by one man's offence death reigned by one; much more they which receive abundance of grace and of the gift of righteousness shall reign in life by one, Jesus Christ.)
18 Therefore as by the offence of one judgment came upon all men to condemnation; even so by the righteousness of one the free gift came upon all men unto justification of life.
19 For as by one man's disobedience many were made sinners, so by the obedience of one shall many be made righteous.
20 Moreover the law entered, that the offence might abound. But where sin abounded, grace did much more abound:
21 That as sin hath reigned unto death, even so might grace reign through righteousness unto eternal life by Jesus Christ our Lord.
I think Gregory Nazianzus states the reason why he had to die (https://www.newadvent.org/fathers/3103a.htm):
For that which He has not assumed He has not healed
If Christ did not die, our death would not have been healed or transformed.