The proof depends on how you're setting things up. In my opinion the cleanest approach is the Lie algebraic one, and it goes as follows. Your Borel subalgebra $\mathfrak b$ determines a choice of simple roots $\Delta$ and consequently a choice of positive roots $\Phi^+$: $\mathfrak b = \mathfrak t \oplus \bigoplus_{\alpha \in \Phi^+} \mathfrak g_\alpha$. The action of $w \in W$ takes $\mathfrak b$ to $\mathfrak b_w = \mathfrak t \oplus \bigoplus_{\alpha \in \Phi^+} \mathfrak g_{w\alpha}$. With respect to the length function defined using $\Delta$, the longest element $w_0$ of $W$ takes $\Phi^+$ to $-\Phi^+$. It follows that $b_{w_0}$ is the Borel subalgebra opposite to $\mathfrak b$.
Apparently this isn't discussed in any of the published literature, even in the numerous exercises for Bourbaki's Chapter VI on root systems in Lie Groups and Lie Algebras. I'm not sure how strong the evidence is for the assertion here that the minimal length is always $h^\vee -2$. Even though it's true for some small rank cases, has it actually been verified for all types? For example, in the respective types $E_6, E_7, E_8$ we have $h = h^\vee = 12, 18, 30$, and in type $F_4$ we have $h=12$ but $h^\vee = 9$.
The underlying question here is whether an element $w$ of minimal length taking a (necessarily long) simple root to $\theta$ can be characterized uniformly in some way. (A further question is how interesting such a result would be: are there are useful consequences if $\ell(w)$ does turn out as predicted?) From a look at small rank examples, I don't yet see any uniform pattern here for the choice of $w$. So it's important to clarify how far the assertion about $\ell(w)$ has been verified case-by-case. Probably the length question might be studied more uniformly in two ways: using just the axiomatics of root systems, or using specific features of the adjoint representation of a simple Lie algebra. But neither approach looks straightforward.
[By the way, the dual Coxeter number has come up in a number of questions on MO. The geometric work by Coxeter treated arbitrary finite real reflection groups, but later it was seen that the order $h$ of a Coxeter element in a crystallographic group (such as the Weyl group of an irreducible root system) can be characterized as 1 plus the height of the highest root. Then it was useful in Kac-Moody theory to define $h^\vee$ to be 1 plus the height of the highest short root if there are two root lengths.]
UPDATE: As the comments below indicate, the length in question is treated by Cellini and Papi (Prop. 7.1 in their 2004 paper in Advances in Math. which I provided a link for) as well as in the 2011 thesis of a student of Papi. I had overlooked the paper, so I was overly pessimistic about finding an explicit discussion in the published literature.
Best Answer
As Koen S points out, the longest element of an irreducible Weyl group is treated in an earlier question (in fact, it comes up in several questions). The question asked here presupposes a standard linear realization of the Weyl group, as occurs in the structure theory of a semisimple Lie algebra over $\mathbb{C}$ for instance. In this context, the standard answer given by Max is formulated as Exercise 5 at the end of Section 13 in my 1972 Springer graduate text. However one arrives at this conclusion, it obviously depends on the classification of irreducible root systems.
On the other hand, the question makes sense for any irreducible finite Coxeter group in its usual realization as a reflection group, and is approached in this spirit (via the Coxeter element) toward the end of Section 3.19 in my 1990 book on reflection groups and Coxeter groups.
P.S. For irreducible Weyl groups, a natural motivation for asking this question involves the criterion for all finite dimensional irreducible representations of the associated simple Lie algebra to be self-dual: If the given highest weight is $\lambda$, the dual has highest weight $-w_0 \lambda$ (where $w_0$ is the longest element of $W$). This always coincides with $\lambda$ iff $-1 \in W$.