I think this is not too hard to see at least when the limit is a strict, ordinary limit. As Mike says, since the connecting functors preserve colimits, colimits in the limit category are computed componentwise. Fix a cardinal $\kappa$ to be named later. I claim that $\mathcal{C}$ is locally $\kappa$-presentable, with generating set given by the set of all objects $K = (K_n)_n \in \mathcal{C}$ whose components $K_n$ are $\kappa$-presentable.

First we check that any object $(C_n)_n \in \mathcal{C}$ is a $\kappa$-filtered colimits of such $K$'s. For each $n$ we have $C_n = \varinjlim_{i \in I_n} K_{i,n}$ with $I_n$ $\kappa$-filtered, assuming that $\kappa$ is at least the sup of the accessibility degrees of the $\mathcal{C}_n$. By some sleight of hand, I think we can assume that $I_n$ is independent of $I$ and the structure maps are levelwise so that $(C_n)_n = (\varinjlim_{i \in I} K_{i,n})_n = \varinjlim_{i \in I} (K_{i,n})$ as desired.

Now we check that the $K$'s are $\kappa$-presentable. Suppose that $(C_n)_n = (\varinjlim_{i \in I} C_{i,n})_n$ is a $\kappa$-filtered colimit. Then

$\mathcal{C}((K_n)_n, (\varinjlim_i C_{i,n})_n) 
= \varprojlim_n \mathcal{C}_n(K_n, \varinjlim_i C_{i,n}) \quad \text{by definition of }\mathcal{C} \\
\qquad \qquad \qquad \quad ~  = \varprojlim_n \varinjlim_i \mathcal{C}_n(K_n, C_{i,n}) \quad \text{because }K_n\text{ is }\kappa\text{-presentable} \\
\qquad \qquad \qquad \quad ~  = \varinjlim_i \varprojlim_n \mathcal{C}_n(K_n, C_{i,n}) \quad \text{assuming that }\kappa \text{ is sufficiently large} \\
\qquad \qquad \qquad \quad ~  = \varinjlim_i \mathcal{C}((K_n)_n, (C_{i,n})_n) \quad \text{by definition of }\mathcal{C}$
as desired.

So how big must $\kappa$ be? It must be at least the sup of the accessibility degrees of the $\mathcal{C}_n$'s and it must be strictly larger than the total number of $\mathcal{C}_n$'s. So e.g. a countable limit of locally $\aleph_1$-presentable categories is locally $\aleph_1$-presentable.