The short answer is no. There is a classification of primitive ideals in the enveloping algebra (and quantised enveloping algebra). This reduces the problem to primitive rings. However the representation theory of primitive rings which are not Artinian is complicated.
An example which I find easier is the Weyl algebra (or linear differential operators). This ring is primitive since the vector space of polynomials is an irreducible faithful representation. This ring is in fact simple (no proper ideal). However the representation theory encompasses the theory of linear differential equations with polynomial coefficients.
So speaking heuristically, the representation theory of semisimple Lie algebras is at least as complicated as the representation theory of the Weyl algebra and it is unreasonable to expect an answer in this case.
I don't know of a formal result that says this is an unreasonable request. For example: does this problem include the problem of classifying indecomposable representations of a wild algebra?
Edit I have just found this reference which solves the question for $sl(2)$.
MR0605353 (83c:17010) Block, Richard E. The irreducible representations of the Lie algebra $sl(2)$ and of the Weyl algebra. Adv. in Math. 39 (1981), no. 1, 69--110.