Quite the contrary, copper(I) is actually relatively stable compared with most fourth-period transition metals in the +1 oxidation state. As discussed here, copper in the +1 oxidation state can be stabilized, even in water solvent, by complexing with a soft base such as thiourea or chloride ion. That does not work with the +1 oxidation state in earlier transition metals in the series.
What happens with most fourth-period transition metals is an imbalance between ionization energy and solvation energy: the second ionization energy is small enough so that the additional solvation energy of a +2 ion versus a +1 ion by water outweighs it, therefore the +1 ion goes on to +2 (or more, especially early in the transition series).
With copper, however, the filled $d$ valence shell whose electrons only partially shield each other from the nuclear charge raises the second ionization energy of the copper. We still favor going to the +2 oxidation state when copper is complexed by water. But a ligand with more covalent bonding character and weaker electrostatic attraction, meaning a soft base, might now allow the +1 oxidation state to remain stabilized given copper's slightly higher second ionization energy. Then we get a stable copper(I) complex with the soft base, which in some cases is also soluble in water.
In the above, copper is contrasted with earlier transition elements in the fourth period. What about a comparison with zinc and later metals in the period? In these cases, the second ionization energy again drops off, until we reach the metalloidal elements arsenic and selenium, because the second ionization no longer comes from the $3d$ subshell (it comes from $4s$ or $4p$). Thus the post-transition metals in Period 4 also tend to go on to a +2 or higher oxidation state while copper, given the right environment, can be kept at +1.