According to Advaita, does the person become Krishna once he attains moksha?
Impossible to become Krishna in Advaita Vedanta or any Vedanta.
In Advaita Moksha, the transformation of the jiva happens to Nirguna Brahman and it is Saguna Brahman who does the transformation. The question of becoming Saguna Brahman (which maps to Krishna in Advaita) is an impossibility not only from the view of Brahma Sutras, furthermore there is no Acharya or scripture that has propounded such a view. Nirguna Brahman is simply the jivatma without any covering of mind or matter i.e. pure self.
From Kanchi Paramacharya's discourses, this is the finale of Advaita Sadhana
Mark it! There is a difference! The One who does the transformation here is the Ishvara who is the saguNa-brahman. But the transformation he does to the JIvAtmA is the formless nirguNa-brahman! And the Jiva does the dhyAna only to become nirguNa and not for becoming the saguNa Ishvara!
The essential identity of the Atman and brahman is the most important tenet of advaita. brahman is the substratum on which all phenomena are experienced, and also the antaryAmin, the One Lord who dwells in all beings. The innermost Atman, the real Self, is the same as this antaryAmin, and identical to brahman. Liberation (moksha) consists in realizing this identity, not just as a matter of literal or intellectual understanding, but as something that is to be grasped by the individual in his/her own personal experience.
According to advaita, technically, there is no person or Krishna. So, given that everything is Brahman and moksha is just realizing this oneness, there is no question of a person becoming Krishna, because they are unreal in the first place. The interpretation of the qualifier "unreal" is of course out of scope of this question.