Identify the appropriate reagent 'X' to carry out the given...

The given transformation involves the selective reduction of an ester group ($-\mathrm{COOCH}_3$) to an aldehyde group ($-\mathrm{CHO}$) without affecting the carbon-carbon double bond ($-$\mathrm{CH}=\mathrm{CH}$-$).

Let us evaluate the given reagents:
$\bullet$ $\mathrm{H}_2, \mathrm{Pd}/\mathrm{BaSO}_4$ is used for the Rosenmund reduction, which converts acid chlorides to aldehydes, but it does not reduce esters.
$\bullet$ $\mathrm{NaBH}_4$ is a mild reducing agent that reduces aldehydes and ketones to alcohols but generally does not reduce esters.
$\bullet$ $\mathrm{LiAlH}_4$ is a strong reducing agent that reduces esters completely to primary alcohols.
$\bullet$ $\mathrm{DIBAL-H}$ (Diisobutylaluminium hydride) followed by hydrolysis selectively reduces esters to aldehydes at low temperatures (typically $-78^\circ\mathrm{C}$) and does not reduce isolated carbon-carbon double bonds.

Therefore, the appropriate reagent 'X' is (i) $\mathrm{DIBAL-H}$, (ii) $\mathrm{H}_2\mathrm{O}$.

Answer: (i) $\mathrm{DIBAL-H}$, (ii) $\mathrm{H}_2\mathrm{O}$