The past in Asgard? No. He has no memory of his past in Asgard as much as he has no memory of his previous life in his canon world. However, he does know what his past self has done in this game before he arrived.

Although, I think you must be referring if he remembers the previous trip before his canon update, then yes he has full memory of it still. He should have all his current Asgardian memories in order now except the memories of the previous incarnation of Soushi with the journal onlyserves.