I believe there is such a description, though I think its pretty debatable whether it is likely to tell you very much about Schubert calculus.

Category $\mathcal{O}$ has a nice collection of objects called "tilting modules"; these are distinguished by have a Verma and dual Verma filtration (actually all of these are self-dual); the indecomposables are indexed by the elements of the Weyl group (look at the lowest elements whose associated Verma or dual Verma appears in the filtration). The self-dual projective $P(w_0)=T(e)$ is an example of a tilting module.

Furthermore, these all have graded lifts in the graded version of category $\mathcal{O}$; in particular, there's a way of grading the Hom spaces between these objects so that the endomorphisms of $P(w_0)$ become $H^*(G/B)$ with the homological grading. If you choose the gradings correctly, the Hom spaces $Hom(T(e),T(w))$ and $Hom(T(w),T(e))$ have lowest degree $\ell(w)$ and dimension 1 in that degree. I believe the Schubert class for $w$ is (up to scalar) the composition of elements from these Hom spaces in lowest degree.

I won't give a detailed proof, but the point is that from Soergel's work you can identify $Hom(T(e),T(w))\cong Ext^\bullet(\mathbb{C}_{G/B},\mathbf{IC}_{S_w})$ and $Hom(T(w),T(e))\cong Ext^\bullet(\mathbf{IC}_{S_w},\mathbb{C}_{G/B})$ with composition being Yoneda product. This shows that any map $T(e)$ to $T(e)$ that factors through $T(w)$ is a sum of Schubert classes for $S_{w'}$ with $w'>w$ (it also shows the claim I made about Hom spaces). Thus, $S_w$ is (up to scalar) the only such element of degree $\ell(w)$.

1. I believe there is such a description, though I think its pretty debatable whether it is likely to tell you very much about Schubert calculus.

Category $\mathcal{O}$ has a nice collection of objects called "tilting modules"; these are distinguished by have a Verma and dual Verma filtration (actually all of these are self-dual); the indecomposables are indexed by the elements of the Weyl group (look at the lowest elements whose associated Verma or dual Verma appears in the filtration). The self-dual projective $P(w_0)=T(e)$ is an example of a tilting module.

Furthermore, these all have graded lifts in the graded version of category $\mathcal{O}$; in particular, there's a way of grading the Hom spaces between these objects so that the endomorphisms of $P(w_0)$ become $H^*(G/B)$ with the homological grading. If you choose the gradings correctly, the Hom spaces $Hom(T(e),T(w))$ and $Hom(T(w),T(e))$ have lowest degree $\ell(w)$ and dimension 1 in that degree. I believe the Schubert class for $w$ is (up to scalar) the composition of elements from these Hom spaces in lowest degree.

I won't give a detailed proof, but the point is that from Soergel's work you can identify $Hom(T(e),T(w))\cong Ext^\bullet(\mathbb{C}_{G/B},\mathbf{IC}_{S_w})$ and $Hom(T(w),T(e))\cong Ext^\bullet(\mathbf{IC}_{S_w},\mathbb{C}_{G/B})$ with composition being Yoneda product. This shows that any map $T(e)$ to $T(e)$ that factors through $T(w)$ is a sum of Schubert classes for $S_{w'}$ with $w'>w$ (it also shows the claim I made about Hom spaces). Thus, $S_w$ is (up to scalar) the only such element of degree $\ell(w)$.

2. Yes, if you're willing to think about $\mathrm{End}(P(w_0))$ via its canonical isomorphism with the center of category $\mathcal{O}$. It's the induced action on the center of a categorical braid group action. See Section 3 of this paper of Stroppel: http://arxiv.org/abs/math.RT/0608234