Here are some details of the solution patched together in the comments. There is an implicit assumption that $\Gamma$ is finitely generated, else it does not have a finite sided fundamental domain. Pick a $\Gamma$-equivariant system of pairwise disjoint horoball neighborhoods $B_\xi$ of the cusps $\xi \in \partial \mathbb{H}^2$.
Suppose that $D$ is a finite-sided fundamental domain for $\Gamma$ (bounded by geodesic paths), for example a Dirichlet domain or a Ford domain. Consider a pointcusp $\xi \in \partial \mathbb{H}^2$$\xi$ on which $D$ accumulates. Since $D$ is finite sided, the only way it can accumulate on $\xi$ is for there to exist a concentric horoball $B$ based at $\xi$$B' \subset B_\xi$ such that $D \cap B$$D \cap B'$ is the region of $B$$B'$ between two rays ending at $\xi$. This induces a decomposition of(taking this concentric horoball is necessary to avoid the parts where $D$ into a unionmucks around inside $B_\xi$ doing unpleasant things close to the boundary of $B_\xi$). After equivariantly shrinking the horoballs, we can assume that $D$ hits each horoball in this standard manner.
Now suppose that $D_1,D_2$ are two piecessuch fundamental domains. After further equivariant shrinking of the horoballs, we can assume that both $D_1$ and $D_2$ hit each horoball in the so-called "thick-thin" decompositionstandard manner. It's now easy to prove that $D_1,D_2$ are comparable: their portions outside the thin part ishoroballs each have compact closure and so are comparable; and if we pick a representative cusp $\xi$ of each $\Gamma$-orbit of cusps, their portions inside $B_{\xi'}$ for cusps $\xi'$ in the $\Gamma$-orbit of $\xi$ translate to a finite union of the horoball pieces; the thick part is everything else, which is compactstandard intersections with $B_{\xi'}$.