How can I Compute the homotopy groups $\pi_n(\mathbb RP^2 \times \mathbb S^3)$ ?
Which theorems will help me in this? Should I use Kunneth theorem?
I have seen this post here Calculate $[\mathbb{S}^n, \mathbb{RP}^{n-1}] \cong \pi_n(\mathbb{RP}^{n-1})/\pi_1(\mathbb{RP}^{n-1})$ but I am not sure how will this help me, if it will, by any means. Any clarification will be greatly appreciated!
It is well known that $\pi_n(X\times Y)$ is isomorphic to $\pi_n(X)\times \pi_n(Y)$.
And so being able to calculate $\pi_n(\mathbb RP^2 \times \mathbb S^3)$ means being able to calculate $\pi_n(\mathbb S^3)$. But AFAIK this still is one of the biggest open problem in homotopy theory. Except for $\mathbb{S}^1$ every sphere has some homotopy groups which we don't know how to calculate. And whatever we known about them is highly complicated, one may say: mathematics of the highest order.
If someone gave you this as an exercise, then there's some major mistake here. You mention Kunneth, so perhaps you were meant to calculate homology instead of homotopy groups? If so, then Kunneth (which applies to homology, not homotopy) is a good approach. Homologies are often easier to calculate then homotopies, especially for such well behaving spaces (CW complex).