As a complement to JVP's answer, here is a direct proof to that $\tilde{H}\cdot C\geq 0$.

Note that nefness is numerically invariant. To check the nefness of $h-e$, we only need to show that for any irreducible curve $C$ in the blowing-up, the intersection number $(h-e)\cdot C$ is nonnegative. If $C$ is not contained in $e$. then the image of $C$, denoted by $D$, is still a curve. In this case, by projection formular, $(h-e)\cdot C=H\cdot D\geq 0$, where $H$ is any hyperplane in $\mathbb{P}^n$. In the case $C$ is contained in $e$, $h\cdot C =0$ however $-e\cdot C =-\deg N_{e/X}|C=1$, where $N_{e/X}$ is the normal bundle of the exceptional divisor in the blowing up $X$. Therefore $h-e$ is nef.

As a complement to JVP's answer, here is a direct proof to that $\tilde{H}\cdot C\geq 0$.

Note that nefness is numerically invariant. To check the nefness of $h-e$, we only need to show that for any irreducible curve $C$ in the blowing-up, the intersection number $(h-e)\cdot C$ is nonnegative. If $C$ is not contained in $e$. then the image of $C$, denoted by $D$, is still a curve. In this case, let $H$ be a hyperplane passing through $P$ but not containing $C$. Then $\tilde{H}$ does not contain $C$ and hence $(h-e)C=\tilde{D}C\geq 0$. If $C$ is contained in $E$, then $h\cdot C =0$. But $-E\cdot C =-\deg N_{E/X}|C\geq 1$, where $N_{E/X}$ is the normal bundle of the exceptional divisor in the blowing up $X$. Therefore $h-e$ is nef.