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 $Y$.