Does my brother, to whom I sold my apartment, have the right to rent it out and benefit from its rent after he bought another house and moved into it, knowing that the agreement between us gives him a grace period for handover until the middle of next year?

Your agreement to exclude the benefit of the apartment for your brother for a known period is permissible, based on Jabir's hadith regarding the stipulation of a riding camel, and because it is permissible to stipulate a known benefit in a sale.

Accordingly, your brother has the right to utilize the apartment for the agreed-upon period. He may use it himself or through another (such as by renting it out), and he may also waive his condition.