Let us consider the arithemetic surface case, which is already very difficult (see the recent work by Gerald Montplet, for example). In this case Faltings-Riemann-Roch established that
$$
\chi(O_{X}(D))-\chi(O_{X})=\frac{1}{2}(D,D-K)
$$
Therefore formally in order to evaluate $\chi(O_{X}(D))$ it is suffice to find $\chi(O_{X})$.

There are however at least two issues at here, one is that the intersection pairing $$
(D, D-K)
$$
in Arakelov theory depends on the choice of Green functions, which is dependent upon the choice of the Arakelov metric. If we use a different metric, then everything will have to be changed. The other issue is Faltings' construction using the technique of "reduction to the Jacobian" in an essential way and cannot be easily generalized to higher dimensions (see the MathScithnet review, for example).

The one-step "magic solution" of the issue essentially comes from the introduction of the Quillen metric, namely finding a metric on the determinant line bundle over $X_{\infty}$ that satisfies the following properties:

1) It should be smooth on the moduli space of genus $g$ complex algebraic curves. Namely, when we "slide" on the "huge" space parametrizing the whole space of genus $g$ complex algebraic curves, the metric on the determinant line bundle should be a smooth function.

2) It should be compatible with Serre duality.

3) It should not be really dependent upon the metric. Namely if we have two different metrics $d_1, d_2$ on $X$, then the metric on the determinant of cohomology should be independent of $s$ for $d_s=sd_1+(1-s)d_2$. In other words, it should be coming from a topological invariant.

4) It should be naturally generalizable to higher dimensions, not using special nice properties only available for Riemann surfaces.

In Ray-Singer's first paper, they proved (3) for analytic torsion (not for Quillen metric, which did not exist by then). For a survey paper on this, see Pavel Mnev's article. In Quillen's paper, he claimed (1) and Soule provided a detailed proof in his book. I think (2) comes from Deligne's introduction of Deligne pairing and (4) comes from Bismut-Gillet-Soule's work. I think this is partly why analytic torsion considered to be so important for Arakelov theory. I do not know any other analytic invariant that can satisfy (1)-(4) in the same time. If it exists, then it has potential to be the building block for a generalized Arakelov cohomology theory.

Part of the difficulty is that analytic torsion is a "secondary global invariant" that involves all positive time of the heat kernel (you have to deal with $Tr(\int^{\infty}_{0}t^{s}e^{-\Delta t}dt|'_{s=0}$). So any potential good candidate that can define $h^{0}_{Ar}, h^{1}_{Ar}$ would have to involve spectral theory of elliptic operators in an essential way. And it is not just the $\zeta$-function of the operator, but its derivative at $s=0$. Therefore the problem become exceedingly difficult and from what I read even computing $\chi(O_{X})$ for $\mathbb{P}^{N}$ in general is not easy.

I am really a beginner in the field and maybe for you discussing this with experts (Gillet, Soule, Bost, Montplet, Faltings, etc) will be helpful.