In a voltaic cell there is zinc and copper and the electrolyte is sodium chloride solution. The question was:
In this cell, which of the following undergoes reduction?
My answer was copper, but why is the exact answer "a hydrogen ion"?
Copper cannot undergo reduction because, in the list of components of the cell you listed, it is only present in its most reduced state, Cu(0). Thus, it cannot be copper that is reduced. See here for additional hints about your system.
You may have been confused by the fact that, in the historical (and widely taught) Daniell cell, copper is indeed reduced: it is present as Cu(+II) in a solution of copper sulfate.