I can't say I understand your notion of "morphism of quiver representations," but I can tell you a canonical bijection between representations for one orientation and another, which might be what you're looking for.
Let $\bar Q$ be the doubled $A_n$ quiver, that is, the same set of dots with all arrows replaced by a pair of oppositely oriented arrows.
Given a representation $M$ of $Q$, consider the representations $E_M$ of $\bar Q$ on the same vector space that agree with $M$ on the arrows in $Q$, do whatever they want on the others, and have the property that the paths $i\to i-1 \to i$ and $i\to i+1\to i$ give the same self-map of $M_i$. Note that this is a set of linear equations on the choices for each new arrow in $\bar Q$, so its set of solutions is a linear space.
Now, consider the map of this linear space to the set of $Q'$-representations given by forgetting the arrows not in $Q'$. This isn't independent of our choices, but there's one isomorphism type that occurs generically in $E_M$ and all the others only show up in lower dimensional subvarieties
So, consider the map $\varphi$ that sends a isomorphism class of $Q$ reps to the $Q'$ rep that occurs generically in $E_M$ after forgetting the arrows not in $Q'$.
Theorem. The map $\varphi$ is a bijection.
Note that this bijection is not extremely easy to describe combinatorially; for example, it is piecewise linear in the multiplicity of the different indecomposables in your representation.
EDIT: Why does this work? Well, by definition, a preprojective lift of a $Q$-rep is an element of the conormal bundle to the $G$-orbit (as usual, $G$ is the product of $GL_n$'s acting on the spaces of the quiver rep) through that point. Since $G$ has finitely many orbits (Dynkin type is very necessary here!), the space of preprojective representations is the union of the conormal bundles to $G$-orbits, and each component is the closure of exactly one of these (this is 14.2 in Lusztig's "Quivers, perverse sheaves and quantized universal enveloping algebras"). Thus, there is a canonical bijection between preprojective components and isomorphism types of $Q$-reps. One map is close up the conormal bundle to the space of reps of that type, the opposite is grab a generic element of the component and forget the arrows that aren't in $Q$.
So the bijection I described is applying one of these maps for $Q$ and the other for $Q'$.