The cool sepulchre is a reference to the Holy Sepulchre, the tomb in which Christ was laid following his crucifixion and from which his body later disappeared. The grave clothes, the garments which had been used to bind his body, were left behind in the tomb. The fact that the grave clothes remained was seen by Christians as evidence for Jesus' Resurrection.
Julia seems to be suggesting that she was more aware of the distressing aspects of Christianity than of its comforting, or redeeming aspects. In particular, she was more aware of the suffering of Christ than of his resurrection.