There is no estimate of the area of the image. I suppose you consider only non-trivial embeddings (inducing non-trivial homomorphisms of the fundamental group). On your cylinder $C_R$ make a cut $[\delta,R]$, where $\delta\in(0,R)$. The resulting region is a ring, and the conformal modulus of this ring changes from $0$ to $R$. Therefore for every $r\in(0,R)$ there is a conformal embedding of $C_r$ into $C_R$. While the area of the image equals the area of $C_R$.

There is no estimate of the area of the image. I suppose you consider non-trivial embeddings (inducing non-trivial homomorphisms of the fundamental group). On your cylinder $C_R$ make a cut $[\delta,R]$, where $\delta\in(0,R)$. The resulting region is a ring, and it is conformally equivalent to $C_{r}$ for some $r=r(\delta)$. This $r(\delta)$ is a continuous function, and we have $r(\delta)\to 0$ as $\delta\to 0,$ and $r(\delta)\to R$ as $\delta\to R$. Therefore for every $r\in(0,R)$ there is a non-trivial conformal embedding of $C_r$ into $C_R$. While the area of the image is equal to the area of $C_R$.

That there is also no estimate for trivial embeddings, is evident.

On conformal moduli read Ahlfors, Conformal invariants, McGrow Hill 1973. There is a recent reprint by AMS.