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In this paper, we prove that a non-projective compact K\"ahler $3$-fold with nef anti-canonical bundle is, up to a finite \'etale cover, one of the following: a manifold with vanishing first Chern class; the product of a K3 surface and the projective line; or a projective space bundle over a $2$-dimensional torus. This result extends Cao-H\"oring's structure theorem for projective manifolds to compact K\"ahler manifolds in dimension $3$. For the proof, we investigate the Minimal Model Program for compact K\"ahler $3$-folds with nef anti-canonical bundles by using the positivity of direct image sheaves, $\mathbb{Q}$-conic bundles, and orbifold vector bundles., Comment: v2; 39 pages; An erroneous argument in Case 1 of Subsection 4.3 has been replaced with a correct alternative proof.; to appear in Mathematische Annalen