  We consider i.i.d. first-passage percolation (FPP) on the two-dimensional
square lattice, in the critical case where edge-weights take the value zero
with probability $1/2$. Critical FPP is unique in that the Euclidean lengths of
geodesics are superlinear, rather than linear, in the distance between their
endpoints. This fact was speculated by Kesten in 1986 but not confirmed until
2019 by Damron and Tang, who showed a lower bound on geodesic length that is
polynomial with degree strictly greater than $1$. In this paper we establish
the first non-trivial upper bound. Namely, we prove that for a large class of
critical edge-weight distributions, the shortest geodesic from the origin to a
box of radius $R$ uses at most $R^{2+\epsilon}\pi_3(R)$ edges with high
probability, for any $\epsilon > 0$. Here $\pi_3(R)$ is the polychromatic 3-arm
probability from classical Bernoulli percolation; upon inserting its
conjectural asymptotic, our bound converts to $R^{4/3 + \epsilon}$. In any
case, it is known that $\pi_3(R) \lesssim R^{-\delta}$ for some $\delta > 0$,
and so our bound gives an exponent strictly less than $2$. In the special case
of Bernoulli($1/2$) edge-weights, we replace the additional factor of
$R^\epsilon$ with a constant and give an expectation bound.
