Let $C,A$ be matrices (not necessarily square) such that $CA$ makes sense.
If I have information about upper bounds on $\|A\|_2$ and $\|CA\|_2$, can I obtain an upper bound on $\|C\|_2$?
Say $\|A\|_2 < k$, $\|CA\|_2 < m$. Do I need more information to find an upper bound for $\|C\|_2$?
No. E.g. if some row of $C$ lies inside the left null space of $A$, you can magnify that row (and hence $\|C\|_2$) indefinitely without affecting $\|CA\|_2$. For a concrete example, consider $$ C=\pmatrix{1&0\\ 0&c},\ A=\pmatrix{1&0\\ 0&0}=CA. $$ We have $\|A\|_2=\|CA\|_2=1$ for all $c$, but $\|C\|_2\to\infty$ when $c\to\infty$. Here the second row of $C$ lies inside the left null space of $A$.
Another example: when $A$ is square, since $\|CA\|_2\ge\|C\|_2\,\sigma_\min(A)$, it is $\sigma_\min(A)$ rather than $\sigma_\max(A)$ that gives useful information. For instance, consider $$ C=\pmatrix{c&0\\ 0&1},\ A=\pmatrix{\frac1c&0\\ 0&1},\ CA=I. $$ When $c>1$, $\|A\|_2=\|CA\|_2=1$, but $\|C\|_2=c$ is unbounded. The norm of $A$ is irrelevant here.