Such an ideal does not exist.
Indeed, suppose the contrary, and let $I$ be such an ideal. The sequence $(x_n)=(1,0,1,0,\dots)$ is almost convergent to $1/2$. So, $$\mathbb N=\{n\in\mathbb N\colon|x_n-1/2|\ge1/2\}\in I, $$ and hence $I$ is the powerset of $\mathbb N$. So, every sequence is almost convergent, to every real limit, which is of course absurd.