What should I do, as I grew up with my mother's family name, and after embracing Islam, I discovered that I must be named after my father's family name, “Al-Masihi” (the Christian), who converted to Christianity before I was born? I do not know if "Al-Masihi" is his real surname or if he changed it, and I have no way of contacting my father's family.

You are not required to change your family name to "Al-Masihy" (The Christian) due to uncertainty about its authenticity and the strong presumption that it is a name altered from the real one. Your father's old apostasy from Islam does not permit him to change the family name, as this is a blameworthy tampering with lineages. The prohibition against attributing oneself to other than one's father is restricted by knowledge and intent; so if you do not know your true lineage, it is sufficient for you to attribute yourself to your father's and grandfather's names without mentioning the family name. If you cannot change the official documents, there is no sin upon you to keep your documents as they are until the matter becomes clear to you, while being careful not to verbally attribute yourself to your mother's family. Islamic Sharia (definition) does not recognize a Muslim's departure from their religion, so the children of a Muslim remain within the fold and ruling of Islam, and if they return to their Islam, they have repented to their original religion.