UPD. Bound simplified.
Here is a constructive bound for the number of solutions to $\phi(x)=m$.
Let $\varphi(a) = m$. If $p^k\mid a$ for some $k\geq 1$, then $p^{k-1}(p-1)\mid m$, and thus $k\leq 1+\frac{\log(m)}{\log(p)}\leq 1+\frac{\log(m)}{\log(2)}$. Then the number of such $a$ is bounded by $$\prod_{d\mid m} (2+\frac{\log(m)}{\log(2)}) = (2+\frac{\log(m)}{\log(2)})^{\tau(m)}.$$
For $m>40$, we have $2+\frac{\log(m)}{\log(2)}\leq 2\log(m)$, and thus we generously bound the number of solutions by $$(2\log(m))^m=2^me^{\log(m)^2}.$$