How is CFL-reachability solvable in exponential time and space?
Please refer to the paper Program Analysis via Graph Reachability by
Thomas Reps where he mentions clearly that CF-reachability is solvable in cubic time.

I assume CFL-reachability is defined as follows:

A CFL-reachability problem is  where a path is considered to connect two nodes only if the concatenation of the labels on the edges of the path is a word in a particular context-free language (an $L$-path):

Definition 2.1 (from paper above). Let $L$ be a context-free language over alphabet $\Sigma$, and let $G$ be a graph whose edges are
labeled with members of $\Sigma$. Each path in $G$ defines a word over $\Sigma$, namely, the word obtained by concatenating, in order, the labels of the edges on the path. A path in $G$ is an $L$-path if its word is a member of
$L$.