Classifying all irreps of an algebra like a W-algebra is a fool's errand, so I'm assuming that's not what you mean.
For finite dimensional irreps, there are 2; for $\mathfrak{sl}_n$, the number is always the number of standard tableaux of the corresponding Young diagram. This is Theorem C in Brundan and Kleshchev's Representations of shifted Yangians and finite W-algebras. This theorem also gives a construction, as does B&K's "Schur-Weyl duality for higher levels".
EDIT: It occurs to me that the W-algebra in this case is actually one of the algebras considered by Musson and van der Bergh in Invariants under tori of rings of differential operators and related topics. Thus, the category of all weight modules with integral regular central character actually has a nice description; it's the quotient of the path algebra of the doubled $A_4$ quiver by the relation the loops going left and going right from either of the two middle vertices are equal. The finite dimensional reps are the two middle vertices. This stuff is discussed (admittedly, without drawing the connection to W-algebras) in our recent paper Hypertoric category $\mathcal O$, especially Example 4.12.