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 finite-dimensional 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 (Borel-Weil 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 non-spin 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 Weyl-type constructions for the rest of the finite-dimensional 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.