Will the father be absolved of sin by spending the 500 EGP he took from his deceased aunt's purse—given that he is not one of her heirs—on repairing the tomb, and if not, what should he do?

If your father spent money on the grave after burial for additional works such as building on the graves, this does not absolve him. If he spent it on preparing his aunt's burial, such as digging the grave, then there is a detailed explanation: if the heirs are absent or minors, and there is no guardian or ruler, and preparing the funeral is an obligation upon Muslims, then whoever undertakes this can claim reimbursement from the estate or from those responsible for maintenance, provided there was an intention to claim reimbursement. Otherwise, he cannot claim reimbursement, and his responsibility is not absolved except by returning the money to the heirs or seeking their permission.