[Note: This answer was updated according to @MarcoGolla's comment to avoid a potential circularity.]
(1) The existence of such a bordism is already answered by your question: "$\mathbb{CP}^2$ generates $\Omega^{SO}_4$" literally means any closed oriented smooth 4-manifold is oriented cobordant to some copies of $\mathbb{CP}^2$ or $\overline{\mathbb{CP}^2}$. So I suppose that you are asking for an explicit bordism.
As @MarcoGolla pointed out, there is a diffeomorphism
$$K3\mathbin\#\mathbb{CP}^2\ \simeq\ 4\mathbb{CP}^2\mathbin\#19\overline{\mathbb{CP}^2}\,,$$
by Exercise 8.3.4(d) in Gompf and Stipsicz's book 4-manifolds and Kirby Calculus. Tracing the connected sums on both sides, we obtain a compact oriented smooth 5-manifold $W_1$ with boundary
$$\partial W_1\ \simeq\ K3\amalg 20\mathbb{CP}^2\amalg 4\overline{\mathbb{CP}^2}\,.$$
Gluing $W_1$ with 4 copies of $\mathbb{CP}^2\times I$ along the boundary, we obtain a compact oriented smooth 5-manifold $W_2$ with boundary
$$\partial W_2\ \simeq\ K3\amalg 16\mathbb{CP}^2\,.$$
Hence $W_2$ is what you want.
In addition to 4-manifolds and Kirby Calculus mentioned above, Scorpan's book The Wild World of 4-manifolds is also a good place to learn about 4-manifolds.
(2) I don't quite understand the question. If you have a connected spin manifold, it induces a spin structure on each connected component of its boundary.