Following the clarification in the comments, I am interpreting the question as follows.  For an effective Weil divisor $N$ on $Y$ such that for some positive integer $\ell$ the effective Weil divisor $\ell N$ is Cartier, if the pullback effective Cartier divisor $\pi^*(\ell N)$ equals $\ell A$ for an effective Cartier divisor $A$ on $X$ (equality as integral effective Cartier divisors, <b>not</b> just as linear equivalence classes of Cartier divisors), does it follow that $N$ is Cartier on $Y$? The answer to that question is <b>no</b>. The following example is a modification of the example in my comment avoiding the mistake identified by Stefano.  

Let $Y$ be the projective cone in $\mathbb{P}^3$ over a smooth plane cubic curve $C\subset \mathbb{P}^2$.  Let $\pi:X\to Y$ be the minimal desingularization.  The linear projection from $Y$ to $C$ gives a regular morphism, $$\rho:X\to C,$$ that is a $\mathbb{P}^1$-bundle.  The morphism $\rho$ maps the exceptional locus $E$ of $\pi$ isomorphically to $C$, and $E$ is a relative hyperplane class for $\rho$. The normal sheaf $\mathcal{O}_X(\underline{E})|_E$ is isomorphic to $\mathcal{O}_{\mathbb{P}^2}(-1)|_C$. 

Let $H\subset C$ be the restriction to $C$ of a hyperplane. Let $D\subset C$ be a degree $3$ effective divisor such that the divisor $H-D$ has finite order $\ell>1$ in the Picard group of $C$, i.e., $\ell H - \ell D$ is the divisor of a rational function $f$ on $C$.  

Let $M\subset Y$, resp. $N \subset Y$, be the cone over $H$, resp. $D$.  Denote by $\widetilde{M}\subset X$, resp. by $\widetilde{N}\subset X$, the strict transform under $\pi$ of $M$, resp. of $N$.  Note that $\ell \widetilde{M}-\ell\widetilde{N}$ equals the divisor of $f\circ \rho$, as does the total pullback under $\pi$ of the Cartier divisor $\ell N-\ell M$.  Thus, the coefficient of $E$ in the pullback of $M$ equals the coefficient of $E$ in the pullback of $N$.

The pullback of $M$ is the strict transform $\widetilde{M}$ plus the exceptional divisor $E$.  One way to see this is to deform $M$ to a hyperplane section of $Y$ that is disjoint from the vertex of the cone.  Since the intersection number of the total transform of $M$ with $E$ equals $0$, it follows that the coefficient of $E$ equals $1$.    

Therefore also $\ell N$ equals the sum of $\ell \widetilde{N}$ plus $\ell E$.  Although $N$ is not Cartier on $Y$, the pullback of $\ell N$ equals $\ell A$ on $X$ for the effective Cartier divisor $A=\widetilde{N}+E$.