The first order condition is $U_1(C(Y),Y)=0$, where $U_i$ means the partial derivative with respect to the $i$th argument.
Now, totally differentiate with respect to $Y$:
$$U_{12}(C(Y),Y)+U_{11}(C(Y),Y)C'(Y)=0.$$
This can be rearranged to give
$$C'(Y)=-\frac{U_{12}(C(Y),Y)}{U_{11}(C(Y),Y)}.$$
Usually, we expect the objective function is concave (i.e. $U_{11}<0$), so the sign of $C'(Y)$ is the same as the sign of $U_{12}(C(Y),Y)$.
Usually, when applying this kind of comparative static trick, the theory tells us what the sign of $U_{12}$ should be. As an example, if a entrepreneur's output as a function of effort and skill is $y(e,s)$, you might think that $y_{12}>0$ (i.e. an increase in effort yields a bigger increase in output for more skilled people). The analagous exercise to the one you are conducting above would then yield $y_{12}>0\implies e'(s)>0$ so more skilled entrepreneurs try harder.
It's hard to know what the appropriate assumption on $U_{12}$ in your example might be because it's not clear why income is in the utility function in the first place and why $U$ isn't just increasing in $c$. Perhaps you have some application in mind that can pin this down.
The envelope theorem would have a different use here. Suppose we found the optimal $C(Y)$ and are interested in the question "evaluated at $C(Y)$, how does an increase in income affect utility"?
The answer is
$$\frac{d U(C(Y),Y)}{dY}=\underbrace{U_1(C(Y),Y)}_{=0}C'(Y)+U_2(C(Y),Y)$$
The envelope theorem tells us that we can ignore the first term (i.e. the indirect effect of $Y$ on $C$) because the first order condition for optimal choice of $C$ (i.e. $U_1=0$) guarantees this effect is zero.
