My understanding of this is very unsophisticated, but perhaps that means that what I can explain is precisely what you want.
To understand $f^*[H] \in H^2(\tilde Y)$, where $H \subset Y = \mathbb{P}^n$ is a hyperplane, it may help to think of $H$ as the zero set of section $s$ of the anticanonical bundle $\mathcal{O}_Y(1)$. Then the zero set of $f^*s$ (essentially just the preimage of $H$) is a divisor representing $f^*[H]$. If $H$ contained the blow-up locus $X$, then the resulting divisor is the sum of the proper transform $P$ of $H$ (the blow-up of $H$ at $X$) and the exceptional set $\tilde X$. If $H$ was transverse to $X$, then the divisor is really just the proper transform of $H$ (which is exactly the pre-image of $H$, H$ in this case), which is the blow-up $\tilde H$ of $H$ at $H \cap X$.
I presume $d = n-m$, the codimension of $X$ in $Y$.
If $D$ is a divisor in $\tilde X$ representing $[D] \in H^2(\tilde X)$, then $j_*[D]$ is the class $[D] \in H^4(\tilde Y)$ that you get by considering $D$ as a cycle in $\tilde Y$. $g^*\xi$ can be represented by a divisor that is the preimage in $\tilde X$ of a hyperplane in $X$. Writing that hyperplane as the intersection of $X$ with a transverse hyperplane $H \subset Y$, we find that $j_*g^*\xi = [\tilde H \cap \tilde X] \in H^4(\tilde Y)$. $\zeta \in H^2(\tilde X)$ corresponds to the normal conormal bundle of $\tilde X$ in $\tilde Y$, so it is the restriction of $-[\tilde X] \in H^2(\tilde Y)$ to $\tilde X$. Therefore $j_*\zeta = -[\tilde X]^2 \in H^4(\tilde Y)$.
To describe $j_*\zeta$ another way, note that since $\mathcal{N} = \mathcal{O}_X(1)^d$, $\tilde X$ is a trivial bundle $X \times \mathbb{P}^{d-1}$. You can get an explicit trivialisation by picking copy of $\mathbb{P}^{d-1} \cong Z \subset Y$ disjoint from $X$: given points $x \in X$ and $z \in Z$, the line from $x$ to $z$ defines an element in the projectivisation of the fibre of $\mathcal{N}$ over $x$. Let $h$ be the projection $\tilde X \to Z$. Then $\mathcal{O}_{\tilde X}(-1) = g^*\mathcal{O}_X(1) + h^*\mathcal{O}_{Z}(-1)$.
$h^*\mathcal{O}_{Z}(1)$ corresponds to a trivial $\mathbb{P}^{d-2}$ subbundle of $\tilde X$. Such a divisor is the intersection of $\tilde X$ with the proper transform $P$ of a hyperplane $H$ containing $X$ and some hyperplane in $Z$. In other words, $-\zeta = g^*\xi -[P \cap \tilde X] \in H^2(\tilde X)$, so $j_*(\zeta + g_*\xi) = [P \cap \tilde X] \in H^4(\tilde Y)$.
Sanity check: $-[\tilde X] + [\tilde H] = [P]$ implies $-[\tilde X]^2 + [\tilde H \cap \tilde X] = [P \cap \tilde X]$, so it adds up.
