Great question. It's a little of both. (We'd have to define what time means for God.) But I suppose we could infer Origen believed Christ's soul - as he believed of all souls - is eternal.
According to his writings, all souls were created by God at some time prior to conception.
Let us inquire whether God, the creator and founder of all things,
created certain of them holy and happy, so that they could admit no
element at all of an opposite kind, and certain others so that they
were made capable both of virtue and vice; or whether we are to
suppose that He created some so as to be altogether incapable of
virtue, and others again altogether incapable of wickedness, but with
the power of abiding only in a state of happiness, and others again
such as to be capable of either condition. (De Principiis)
And that Christ has one.
Therefore, when the Son of God wished to appear to men and live among
men for the salvation of the human race, He took not only a human
body, as some suppose, but also a soul, and one like our souls in its
nature, but like Himself in purpose and power, and such as could
fulfill without turning all the wishes and dispensations of the Word
and Wisdom. (De Principiis, emphasis mine.)
And all souls are eternal.
The soul, having a substance and life of its own, shall after its
departure from the world, be rewarded according to its deserts, being
destined to obtain either an inheritance of eternal life and
blessedness, if its actions shall have procured this for it, or to be
delivered up to eternal fire and punishments, if the guilt of its
crimes shall have brought it down to this. ( Ante-Nicene Fathers re: Origen, Vol. 4, 1995, p. 240)