I thought that EkhristosAnesti addressed this very well in another response:
I've come to learn that the existence of paradox is the quintessential standard by which the Orthodoxy of any given theory/principle is measured. The pursuit to resolve paradox is in turn the hallmark of heresy.
Or, to quote some Fathers
"We have learned that there is a difference between begetting and procession, but the nature of the difference we in no wise understand." -- St. John of Damascus
"You ask what is the procession of the Holy Spirit? Do you tell me first what is the unbegottenness of the Father, and I will then explain to you the physiology of the generation of the Son, and the procession of the Spirit, and we shall both of us be stricken with madness for prying into the mystery of God."
I've noticed a tendency among many Western Christians to think that in defining the Trinity as 3 Hypostasis with 1 Ousia, the Fathers had somehow logically resolved the paradox that God is 1 and God is 3 and to move from there to attempts to understand the difference between Ousia and Hypostasis in logical terms (i.e., your question). However, for the Nicean-era Fathers, Hypostasis and Ousia were synonyms. The Hypostis/Ousia distinction was useful for defending the revelation of the Trinity not because it explained away the paradox but because it enshrined it. The Fathers could have as easily reversed the terms and it would have meant the same thing--God is distinctly 3 and He is indivisibly 1 and the how 1=3 is not something we can understand. The 3 properties (Ungenerate, Begotton, Proceeding) belong to the 3 Hypostasis because it has been revealed to us that the Father is without beginning, the Son is Begotten of the Father, and the Spirit proceeds from the Father, so just as we know there are 3 persons, we know each possesses one of these properties. But that is something that is revealed to us, not something any Father reasoned out.