Michael Hardy answered the question you seemingly asked: "I don't know how to get one of marginal functions" but as you discovered here, the marginal density of $Y$ seems to be a special function. On the other hand, computing the moments of $Y$ (which is what really you want to do) does not require that you first find the marginal density of $Y$. We have
$$
\begin{align*}
E[Y^n] &= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
y^n f(x,y)\ \mathrm dy\ \mathrm dx\
&= \int_{x = 0}^{\infty}\int_{y = -x}^{y = x} y^n \ \mathrm dy \frac{1}{2x} \exp(-x)\mathrm dx\
&= \int_{x = 0}^{\infty} \left . \frac{y^{n+1}}{n+1}\right\vert_{-x}^x
\frac{1}{2x} \exp(-x)\mathrm dx\
&= \begin{cases}
\int_{x = 0}^{\infty} \frac{1}{n+1}
x^n \exp(-x)\mathrm dx = \frac{\Gamma(n+1)}{n+1}, & n ~\text{even},\
\quad & \
\quad & \
0, & n ~\text{odd},
\end{cases}\
\
\end{align*}
$$
from which you can get the variance of $Y$ and the covariance of $X$ and $Y$.