 Solution

True. Write $C={\bigcup }_{n\ge 1}{C}_n$ with every $C_n$ nowhere dense in $\mathbb{R}$. ThenThe closure of $C_n\times \mathbb{R}$ is
$\overline{C_n}\times \mathbb{R}$, which has empty interior because $\overline{C_n}$ has empty interior. Each product is therefore nowhere dense, so $A$ is meagre in ${\mathbb{R}}^2$.