The paper "On real homotopy properties of complete intersections" by Babenko gives an answer to your question. I will adapt Corollary 1 presented therein to the case of hypersurfaces:

Let $X$ be a degree $d$ hypersurface in $\mathbb{C}\mathbb{P}^{n+1}$, and denote its topological Euler characteristic by $\chi$ (note that $\chi$ depends only on $d$ and $n$). If $\chi = n+1$, then the only rationally non-zero homotopy groups of $X$ are $\pi_2 (X) \otimes \mathbb{Q} = \pi_{2n+1} (X) \otimes \mathbb{Q} \cong \mathbb{Q}$. If $\chi \neq n+1$, then for any $j\geq 1$ we have $$\dim\pi_{j+1}(X)\otimes \mathbb{Q} = \frac{(-1)^j}{j} \sum_{d|j, \ \ d\geq 1} (-1)^d \mu\bigl(\frac{j}{d}\bigr)\sum_{\alpha=1}^{2n-2} \xi_\alpha^{-d},$$

where $\xi_\alpha$ are the roots of $1-(-1)^n(\chi-n-1)(1+z)z^{n-1} + z^{2n-1}$ that are not the root $-1$, and $\mu$ is the Möbius function.

For example, in the case of a $K3$ surface, $n=3$ and $\chi = 24$, and we get $\pi_3(K3)\otimes \mathbb{Q} \cong \mathbb{Q}^{252}$, $\pi_4(K3)\otimes \mathbb{Q} \cong \mathbb{Q}^{3520}$, $\pi_5(K3)\otimes \mathbb{Q} \cong \mathbb{Q}^{57960}$, $\pi_6(K3)\otimes \mathbb{Q} \cong \mathbb{Q}^{1020096}$, $\pi_7(K3)\otimes \mathbb{Q} \cong \mathbb{Q}^{18664800}$, $\pi_8(K3)\otimes \mathbb{Q} \cong \mathbb{Q}^{351204480}$, $\ldots$. This is consistent with Corollary 4.10 in Samik Basu and Somnath Basu's Homotopy groups and periodic geodesics of closed 4-manifolds and the answers to this question What are the higher homotopy groups of a K3 suface?

For convenience, here is a formula for the topological Euler characteristic of a degree $d$ hypersurface in $\mathbb{C}\mathbb{P}^{n+1}$ (simplified thanks to abx's comment); $$\chi = \frac{1}{d}((1-d)^{n+2} + d(n+2) -1).$$