Compound Distribution — Normal Distribution with Log Normally Distributed Variance
Hint:
$X \mid Y \sim \mathcal{N}\left(\mu_{X}, Y\right)$, no?
So $$f_{X}(x) = \int_{-\infty}^{\infty}f_{X\mid Y}(x \mid y)f_{Y}(y)\text{ d}y$$
$F_{X}$ can be easily found from this.
In general, $$\mathbb{E}[g(X)] = \mathbb{E}\left[\mathbb{E}[g(X) \mid Y]\right] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}g(x)f_{X \mid Y}(x \mid y)f_{Y}(y)\text{ d}x\text{ d}y$$