Is it permissible to lend a brother a portion of money to complete the purchase of an apartment, and to give him the other portion from zakat funds, by advancing the zakat for the new year, knowing that the money is deposited in the bank?

It is not permissible to give zakat to a person who possesses an amount that enables him to purchase a dwelling, as he is not among those eligible to receive zakat. You can lend him the remaining amount for the apartment's price, for a good loan (Qard Hasan) carries immense reward. The recipients of zakat have been specified by Allah in the Holy Quran in Surat At-Tawbah (verse 60), and whoever is not among them has no right to zakat.