[Note: This answer was updated according to @MarcoGolla's comment to avoid a potential circularity.]

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**(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*][1] is also a good place to learn about 4-manifolds.

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**(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.

[1]: https://people.math.ethz.ch/~dkosanovic/24-FS/Scorpan-ww4m.pdf