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 $N=1000$)
Now the desired number $p$ of fortunate guests should follow by solving $s_{p,N}=1/N$ for $p$. As a check, for $N=1000$ the exact integer result is $p=94$, while the large-$N$ asymptotics gives $p=94.334$.
I would have guessed $p\approx \sqrt{N}$, but numerically I find $p\approx N^\alpha$ with $\alpha\approx 0.57$ distinctly greater than $1/2$.