Firstly, note that we cannot expect the relationship between the plurigenera of $X$ and those of $J$ to be particularly easy. For example, passing from $X$ to $J$ can change the Kodaira dimension: If $X$ is an Enriques surface, then for any choice of an elliptic fibration on $X$, its Jacobian $J$ is a rational surface. This is because the elliptic fibration on an Enriques surface always has multiple fibers and passing from $X$ to $J$ gets rid of them. But in some sense, multiple fibers are really the main culprit here. In particular, the Kodaira dimension does not change if $X \to B$ does not have multiple fibers. Moreover, even in the presence of multiple fibers, we always have $\kappa(X) \geq \kappa(J)$.
Despite this, we can quite accurately say what the Enriques invariants of $J$ are, given only the Enriques invariants of $X$:
For the Betti numbers we always have $b_i(X) = b_i(J)$. This is Corollary 5.3.5 in the book "Enriques Surfaces I" by Cossec and Dolgachev. In general, Chapter 5 of that book is a good reference for these kinds of things.
For the plurigenera, let us first look at what happens in the Jacobian case. If $J$ is a Jacobian elliptic surface over the base curve $B$ of genus $g_B$, the canonical bundle of $J$ is given as $2g_B - 2 + \chi(\mathcal{O}_J)$ times the fiber class (at least up to algebraic equivalence). Consequently, we see that $\omega_J$ is the pullback of a line bundle $\mathcal{L}$ on $B$ of degree $2g_B - 2 + \chi(\mathcal{O}_J)$. With this notation, we have that $P_n(J) = h^0(\mathcal{L}^{\otimes n})$ and the latter we can compute by Riemann-Roch. Thus, if $\mathcal{L}$ has positive degree and $J$ is not a product ($J$ is a product iff $\mathcal{L} \cong \omega_B$), we have $P_n(J) = n(2g_B-2+\chi(\mathcal{O}_J))+1-g_B$. If $\mathcal{L}$ has negative degree, we have $P_n(J) = 0$ for all $n$. The degree zero case can be dealt with using the classification of surfaces; essentially we see that either $J$ is either a K3 surface or an abelian surface or a bielliptic surface. (We have $g_B = 0$ in the first case and $g_B = 1$ in the other two. Moreover, $J$ being an abelian surface happens only it is a product). In any case, the plurigenera of these are well-known.
The point of all this was that the previous paragraph shows that we can essentially recover the plurigenera $P_n(J)$ simply from knowing $\chi(\mathcal{O}_J)$ and $g_B$. Now, $g_B$ clearly does not change when passing from $X$ to $J$ and neither does $\chi(\mathcal{O}_X)$ (this is in Chapter 5 of Cossec-Dolgachev). As $\chi(\mathcal{O}_X) = 1 - q(X) + P_1(X)$ by Serre duality, this shows that (modulo some hiccups regarding the case when $J$ is a product) we can completely recover the $P_n(J)$ given only $q(X)$ and $P_n(X)$ (unless $J$ is a product, we have $g_B = q(X)$).