Number of Geodesic Paths Passing Through a Vertex in an Expander Graph

My short answer: I think not much is known. But: here is the state of the art on related problems, as far as I am aware.

Aldous and Bhamidi consider the following model. Place independent exponential edge weights on the edges of the complete graph $K_n$; we view the weights as edge lengths. Then, for each pair $u,v$ of vertices, place a constant flow between them on the shortest path from $u$ to $v$. For each edge $e$, write $F_n(e)$ for the total flow along edge $e$ in the resulting network. Then for each fixed $z > 0$, as $n \to \infty$, \[ \frac{1}{n} \#\{e: F_n(e) > z\log n\} \to \int_0^{\infty}\mathbb{P}(W_1W_2e^{-u} > z)~du, \] where $W_1$ and $W_2$ are independent exponentials. (In fact, the paper proves a more detailed distributional convergence result.) In particular, typical edge congestion is $O(\log n)$; but the paper does not address maximum edge congestion. The paper also addresses vertex congestion (more directly linked to your question), showing a similar convergence result but with a more complicated term on the right-hand side of the convergence. As in the edge case, however, maximum vertex congestion is not treated, only typical congestion.

What follows is less relevant as it relates to global strategies for minimizing congestion, rather than greedy routing between nodes. I'm posting it anyway in case it's useful.

Alan Frieze has a survey on disjoint paths in expander graphs which may be of interest. Theorems 4 and 5 of that survey are results of Broder, Frieze and Upfal, which imply that in an expander, any set of at most $c n/\log^2 n$ pairs of vertices can be connected by disjoint paths, and that every pair can be connected by a path in such a way that the total congestion is $O(n \log n)$.

Finally, for a particular class of random expanders, something can be said about a fractional version (flows rather than paths; again, this allows global optimization). Given a connected, undirected graph $G=(V,E)$, a uniform flow of volume $\phi$ on $G$ is a collection $F$ of flows, one for each ordered pair $(v,w)$ of vertices of $G$, each having volume $\phi$. Given $f \in F$ and $e \in E$, write $f(e)$ for the flow through edge $e$ in $f$ (ignore direction so this is always non-negative). Then write \[ \chi(F)=\max_{e \in E} \sum_{f \in F} f(e) \] for the maximum flow across any edge of $G$, when all flows of $F$ are simultaneously active

Aldous, Mcdiarmid, and Scott have proved the following. Fix a non-negative random variable $C$ with $\mathbb{E}(C) < \infty$, and take G_n to be the complete graph $K_n$ each of whose edges $e$ is weighted with an independent copy $C_e$ of $C$. Let $\phi_n$ be the largest value such that there exists a uniform flow $F$ of volume $\phi_n$ on $G$ such that \[ \sum_{f \in F} f(e) \le C_e \] for all $e \in E(K_n)$. Then there is a positive constant $\phi_*$ such that $\phi_n \to \phi_*$ in probability.

Note that if $C$ takes some fixed value $M$ with probability $p$, and is $0$ with probability $1-p$, this is equivalent to requiring maximum congestion $\le M$ on the random graph $G_{n,p}$. Thus, this setting includes (edge) congestion-type bounds on at least some expander-like graphs.