Is there a better equation than the Nernst equation for low concentrations?
The Nernst equation does not need a replacement for low concentrations. Yes the limit is $\infty$, which essentially means you can't realistically have zero concentration of anything. We may think of absolutely pure water as an idealized situation. Well, as soon as it gets in touch with copper, some $\ce{Cu^2+}$ ions instantly jump into the solution, and from that point on, you have a certain non-zero concentration of them (though probably pretty low), and the equation works just fine.