This is more generally true for any representation-finite string algebra (provided as Dag Madsen pointed out you are talking about irreducible maps between indecomposable modules, otherwise every non-projective module is a cokernel of an irreducible monomorphism). String algebras are a subclass of the class of special biserial algebras, where the admissible ideal of the quiver is spanned by zero relations.
For this class of algebras, all indecomposable modules are given by strings (i.e. words in the arrows of the quiver and their inverses such that there is no cancellation). This is essentially due to Gelfand and Ponomarev, see
I. M. Gel'fand and V. A. Ponomarev, MR0229751 Indecomposable representations of the Lorentz group., Russian Mathematical Surveys 23 (1968), no. 2, 3--60. The general case is is stated e.g. in Burkhard Wald and Josef Waschbüsch, MR 801283 Tame biserial algebras, J. Algebra 95 (1985), no. 2, 480--500.
In this case the irreducible monomorphisms are given by appending hooks to a given string. This is proven in M. C. R. Butler and Claus Michael Ringel, MR 876976 Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 15 (1987), no. 1-2, 145--179. Thus, the cokernel of an irreducible monomorphism is just the end of a hook, hence has simple socle.