Let $U_k=X_k-1$. Then $(U_k)_k$ is i.i.d. and centered and, for large values of $n$, $S_n-n=\sum\limits_{k=1}^nU_k$ is approximately $\sqrt{n}$ times a centered gaussian with variance $\mathrm E(U_1^2)=1$. As such, the central limit theorem yields that, for every nonnegative $x$, $\mathrm P(Y_n\geqslant x\sqrt{n})\to\mathrm P(W_1\geqslant x)$ when $n\to\infty$, where $W_1$ denotes a standard gaussian random variable.
Likewise, the functional central limit theorem asserts that the path $(W_n(t))_{0\leqslant t\leqslant 1}$ behaves more and more like the path of a standard Brownian motion $(W_t)_{0\leqslant t\leqslant 1}$. Here $W_n(k/n)=(S_k-k)/\sqrt{n}$ for every integer $0\leqslant k\leqslant n$ and $(W_n(t))_{0\leqslant t\leqslant 1}$ is the linear interpolation of these values.
In particular $\mathrm P(Z_n\geqslant x\sqrt{n})\to\mathrm P(\tau_x\leqslant 1)$ when $n\to\infty$, where $\tau_x=\inf\{t\geqslant0\ ;\, |W_t|\geqslant x\}$.
The distribution of $\tau_x$ is well known and best described by its Laplace transform which is, if I remember correctly, $ \mathrm E_0(\mathrm e^{-\lambda\tau_x})=1/\cosh(x\sqrt{2\lambda}), $ from which the density of $\tau_x$ may be deduced. For a reference, I would check these lecture notes by Yuval Peres and Peter Mörters or one of Rick Durrett's textbooks.