I know explicit realization of irreducible representations of simple lie algebra $sl_n$ when the highest weight of that representation is a fundamental weight.Is there any explicit realization of any irreducible representation when the highest weight is not a fundamental weight?
It depends on how explicit. Abstractly, any finitedimensional representation is the quotient of a tensor product of fundamental representations, or can be realized as sections of a line bundle over a flag variety (BorelWeil construction). If we want really explicit, then we can say more specific things in types ABCDG: For $sl_n$, see Section 15.3 of the book Representation Theory by Fulton and Harris. For the symplectic Lie algebra and all nonspin representations of the orthogonal Lie algebra, see Section 17.3 and 19.5. The basic spin representations are in Chapter 20, and one can use this and 19.5 to get Weyltype constructions for the rest of the finitedimensional representations, see Section 3 of my paper with Weyman http://arxiv.org/abs/1303.0546 for a start on that (the presentation is given, but not worked out in detail). The exceptional types are also discussed in that paper, but only $G_2$ has a complete answer. 

