What does "percentage by mass" mean in the context of a compound dissolved in a solution?
is the definition of percentage by mass different than what I'm using?


Basically, the questioner wants you to calculate mass by volume percentage contrary to your assumption that the question asks for a mass by mass percentage. The existence of a lot of terms referring to the same mass percentage can be a common confusion. 

Nevertheless, as a rule of thumb, you'd say, I advise you to go for the calculation of mass by volume percentage unless the questioner explicitly mentions or you've been given enough data.


  Find the percentage by mass of copper in the leach solution.


Now that you already have the concentration of $\ce{Cu^2+}$ ions with you, you just need to calculate the number of moles, then the mass and finally divide it with the volume of the mixture.


  Also, how does $\ce{H2SO4}$ react with copper compounds to form $\ce{CuSO4}$? I thought copper couldn't reduce hydrogen ions.


Oxidic sources of copper are well known for their amenability to sulfuric acid leaching, for eg. azurite, malachite, tenorite, cuprite, chryosocolla, and brochantite dissolve easily in dilute sulfuric acid. Note that too dilute concentration is refrained from since we don't want hydrous ferric oxides to precipitate (they can adversely effect copper extraction). 

And as a side-note, if you remove copper sulfides and iron sulfides, the $\ce{H2SO4}$ requirement would be about five times the weight of the dissolved copper (concentrated $\ce{H2SO4}$ is a good oxidizer).



Interestingly, this question was asked in University of Cambridge international examinations October/November 2010, your question is perhaps a simplification of this: