The question is natural but looks difficult to approach strictly within the combinatorial definitions. Maybe it's helpful here to suggest a broader geometric framework, which applies more generally to reductive algebraic groups but involves here a general linear group having the symmetric group $S_n$ as Weyl group. Recall that the "Bruhat order" was defined by Chevalley in the more general setting as the order on Bruhat cells in the flag variety given by inclusion of one cell in the closure of another. These cells are naturally labelled by elements of $S_n$.
On the other hand, the associated geometry of the flag variety and its cotangent bundle (work of Springer, Steinberg, Spaltenstein) leads to Springer fibers with equidimensional components. For the general linear case, these Springer fibers in the flag variety correspond to partitions (Jordan forms of unipotent elements). In this set-up, pairs of typical flags in components of a Springer fiber relate naturally to pairs of standard tableaux of the shape determined by an element of $S_n$, correlating with the Robinson-Schensted correspondence. This is brought out by Steinberg in a short paper: An occurrence of the Robinson-Schensted correspondence, J. of Algebra 113 (1988), 523-528. The Bruhat order is lurking in all of this machinery, but the connection with Robinson-Schensted isn't made explicit enough.
A further related paper by Anna Melnikov appears in the Electronic J. of Combinatorics 12 (2005), treating the combinatorics of orbital varieties (but only in a very special case).