The $p$-th guest receives a share of 
$$s_{p,N}=p N^{-p-1} (N-p+1) \frac{\Gamma(N+1)}{\Gamma(N-p+2)}$$
For large $N$ I substitute the asymptotic expansion of the Gamma function,
$$\Gamma(z)\approx e^{-z}z^z(2\pi/z)^{1/2},$$ 
to arrive at the approximation
$$s_{p,N}\approx p(N-p+1) e^{1-p} \sqrt{\frac{1}{N+1}}\sqrt{\frac{1}{N-p+2}} (N+1)^{N+1}  N^{-p-1}  (N-p+2)^{-N+p-1}$$
which agrees very well with the exact value (compare the dots with the continuous curve in the plot below, for $N=1000$)

<IMG SRC="https://ilorentz.org/beenakker/MO/pochhammer.png"/>

Now the desired number $p$ of fortunate guests should follow by solving $s_{p,N}=1/N$ for $p$. I expect an answer $p\approx \sqrt{N}$, but will need to work a bit more for a precise result.