First, since $\overline N$ is contractible its equivariant cohomology is the same as for $pt$. The Poincare pairing is uniquely determined by $\int_{\overline N} 1$ (since it is linear with respect to $H^*_G(pt)$-linear). H^*_G(pt)$).
More precisely, any cohomology class of $\overline N$ has the form $\alpha\cdot 1$ where $\alpha$ is an equivariant cohomology class of $pt$ and $1$ denotes the unit cohomology class in $\overline N$ and we have $$ \langle \alpha\cdot 1,\beta\cdot 1\rangle =\alpha\beta\int_{\overline N} 1. $$
I don't know a good way to compute this $\int_{\overline N} 1$ for arbitrary $N$ - other than replacing $\overline N$ by a resolution and using fixed point localization.
By the way, if $\overline N$ is the minimal orbit, then ${\mathbb C}^2\times {\overline N}$ is the same as the Uhlenbeck space of $\mathbb C^2$ of second Chern class 1 - that should give you another way to compute that integral (is it obvious that you get the same answer?)
