Good question! The reaction written as above is always a net reaction. In the net reaction its easy to see that Cu2+ has got reduced to Cu metal. Indicating it picked up two electrons. Hydrogen started from oxidation state 0 and ends as +1 - losing one electron per atom of hydrogen. So the net reaction tells you who is the terminal reducing agent and oxidising agent. However the real source of electrons that have come to Cu is hard to find/explain.
Such a description for source of electrons has to come from a detailed and well proved mechanism. Otherwise its impossible to trace the movement of electrons. In the reaction presented above which proceeds without the 'need for a catalyst' it may be a good guess that Hydrogen itself donates electrons and does not involve Oxygen for electron transfer.
However, we can disprove that O2- donates electrons, by realising that water also can act as a source of O2-, and a mixture of, say, Copper sulphate, water and hydrogen does not give rise to Copper metal! That is - no redox reaction occurs.
However if a catalyst is involved, it is safe to say, that the catalyst donates electrons to Cu2+ and then the oxidised catalyst is reduced by hydrogen - thereby oxidising Hydrogen(acts as a reducing agents- means it gets oxidised) and restoring the catalyst to its original state. A 'catalyst' is a catalyst because it gets regenerated and jumps back into the reaction cycle ! Net reaction doesnt show involvement of catalyst, doesnt show consumption and regeneration of catalysts. But only shows terminal oxidants and reduction. These sort of examples are in plenty in the realm of organic chemistry and not so much in inorganic chemistry. Ofcourse its attributed to the fact that it is easier to study organic reaction mechanisms compared to inorganic reaction mechanisms. Which in turn is due to the fact that organic reactions occur at reasonable temperatures and rates of reaction are slower than their inorganic counterparts.