The verb in the Aramaic is in the hithpa’al stem, which is reflexive/passive, and so has the idea of “laid up” as noted by the OP. (The relative pronoun daleth “ד” is also prefixed to the verb.) So the verse would read as follows with the verb translated in bold italics:
4 And the Lord said to Moses, “Behold, I will bring down for you bread from heaven, which has been reserved for you from the beginning. And the people will go out and gather a day’s portion every day, in order to test them, [to see] whether they observe the precepts of my Law or not.”
The footnote for this verse in the English translation provides the following:
The manna was among the ten things which, according to rabbinic tradition, were
created on the eve of the first sabbath.
Thus the manna from heaven was stored (“reserved”) in heaven since the first Sabbath (“from the beginning”), and only given to the Israelites when they were traveling in the desert with Moses. The exact same verb form occurs again several verses later, and the translation is the same with the translated verb noted in bold italics:
15 When the children of Israel saw [it], they were amazed, and they said to one another, “What is it?” For they did not know what it was. And Moses said to them, “It is the bread that was reserved for you from the beginning in the heavens on high; and now the Lord is giving it to you to eat.
English translations taken from:
Cathcart, K., Maher, M., & McNamara, M. (Eds.). (1994). The Aramaic Bible: Targum Neofiti 1: Exodus and Targum Pseudo-Jonathan: Exodus. (M. McNamara, M. Maher, & R. Hayward, Trans.) (Vol. 2, Ex 16:4). Collegeville: Liturgical Press.