Is it permissible to act upon the will that designated the eldest sister to receive her mother's gold and one of her parents' apartments, with the approval of both parents and both brothers, or must the entire inheritance be divided after their passing?

What was mentioned regarding the gold is a bequest to an heir, and it is not permissible except under certain conditions. What was mentioned regarding the apartments has details: your brothers' apartments are a gift if they took possession of them before the death of the giver, otherwise the gift is void according to the majority of scholars. As for your apartment, it is a bequest, not a gift. A father must be just in giving gifts to his children, and it is not sufficient for him to bequeath an apartment to you just because you are an heir. If the giver dies before being just, the gift proceeds upon his death, and the recipient must return it to be divided with the inheritance, which is what Ibn Uthaymeen preferred.