What is the ruling regarding the apartment and money owned by an unmarried uncle, 60 years of age, which are not legally registered in his name, in the event of his death, and what must he do so that he bears no sin?

If the uncle's writing to his sister or opening an account for your father was a formality, then it has no effect. However, if it was a will [to take effect] after death, and it was within the limit of one-third [of the estate], it is permissible because they are not heirs. What exceeds one-third depends on the heirs' approval. He is allowed to donate his money during his lifetime, unless he intends to deprive the heirs.