In this paper, we study the following Kirchhoff-type equation:$ \begin{equation*} M\left(\displaystyle{\int}_{\mathbb{R}^2}(|\nabla u|^2 +u^2)\mathrm{d} x\right)(-\Delta u+u) + \mu V(x)u = K(x) f(u) \ \ \mathrm{in} \ \ \mathbb{R}^2, \end{equation*} $where $ M \in C(\mathbb{R}^+, \mathbb{R}^+) $ is a general function, $ V \geq 0 $ and its zero set may have several disjoint connected components, $ \mu > 0 $ is a parameter, $ K $ is permitted to be unbounded above, and $ f $ has exponential critical growth. By using the truncation technique and developing some approaches to deal with Kirchhoff-type equations with critical growth in the whole space, we get the existence and concentration behavior of solutions. The results are new even for the case $ M \equiv 1 $.