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 $f(N)$ of fortunate guests should follow by solving $s_{p,N}=1/N$ for $p\equiv f$. As a check, for $N=1000$ the exact integer result is $f=94$, while the large-$N$ asymptotics gives $f=94.334$.

I would have guessed $f\approx \sqrt{N}$, but numerically I find a slightly more rapid increase, the plot below is a comparison with $f=\sqrt{2N}\log\log N$. In any case, the large-$N$ limit of $f/N$ seems to be zero.

<IMG SRC="https://ilorentz.org/beenakker/MO/fortunate.png" WIDTH="400"/>

*Gold curve: Number $f$ of fortunate guests versus $N$, calculated form the asymptotics of $s_{p,N}$. The blue curve is $f=\sqrt{2N}\log\log N$.*