In practice, given a stress-energy tensor $T_{\mu\nu}$, we may attempt to find solutions to the Einstein field equations using perturbation theory. The basic idea is to expand around a known solution $g_{\mu\nu}$ by a perturbation $h_{\mu\nu}$. In the case of a flat background,
$$\delta G_{\mu\nu} = 8\pi G \delta T_{\mu\nu} = \partial_\mu \partial_\nu h - \partial_\mu \partial_\alpha h^\alpha_\nu -\partial_\nu \partial_\alpha h^\alpha_\mu + \partial_\alpha \partial^\alpha h_{\mu\nu} - \eta_{\mu\nu} \partial_\alpha \partial^\alpha h + \eta_{\mu\nu} \partial_\alpha \partial_\beta h^{\alpha \beta}.$$
In some cases, one may solve the equations exactly, or more typically, employ numerical methods. Another alternative approach when faced with a stress-energy tensor is to try to determine the symmetries the metric may have, and then plug in an ansatz for the metric to yield a set of differential equations which may be more tractable, analytically and numerically.
Yet another alternative is to generate new solutions from old ones through the introduction of pseudopotentials, a method due to Harrison, Eastbrook and Wahlquist, called the method of prolongation structures.
There are several other methods such as those that rely on Lie point symmetries of differential equations. There is also a Backlund transformation, which relies on identifying a simpler differential equation whose solution satisfies a condition involving the solution to the harder problem.
These methods are too involved to present here and require a significant background. They are explained in Exact Solutions to the Einstein Field Equations by H. Stephani et al.
Addressing your other question, if given a metric $g_{\mu\nu}$, of course one can compute $T_{\mu\nu}$ through the Einstein field equations, you just plug it in and tediously compute all the curvature tensors. There are strictly speaking probably some requirements on the functions in $g_{\mu\nu}$, but you can get away with most things, even distributions, such as a delta function, which may lead to a stress-energy tensor describing a brane.
When Einstein derived his equations the Newtonian limit (and it may be taken also in the present of sources) was an important check for the theory. It is also useful to understand the limits of the Newtonian theory, and sometimes helps to work out the intuition about the relativistic case.
However to solve Einstein equations the Newtonian limit is not required and sometimes is not useful at all.
First, there are exact solutions of the Einstein equations: various black holes and the sufficiently symmetric stars, the cosmological solutions and some other situations.
Second, you may use the perturbation theory near the exact solutions. The simplest such application is linearized gravity that describes gravitational waves on a flat Minkowski background. Note that contrary to your original post, this weak gravity limit is not Newtonian. The gravitational waves do not exist in the Newtonian theory.
Third, you may use the Newtonian limit and start with the Poisson equations. Then you may perturb this solution adding relativistic effects. This is,obviously, a method depending on the Newtonian theory. It is known as a post-Newtonian expansion and is different from a weak gravity limit.
You may also solve the Einstein equations numerically. This is e.g. is used to study the black hole collisions where different perturbation theories fail. This is also a situation where the Newtonian theory is useless.
Best Answer
In a $1$-dimensional manifold the Riemann tensor's only component is $R_{0000}=0$ by the tensor's symmetries, so $R_{\mu\nu}=0,\,R=0$. The EFE simplifies to $\Lambda g_{00}=\kappa T_{00}$. The two sides are still not scalar-valued, as they depend on the chosen coordinate system viz. $ds^2=g_{00}(dx^0)^2$.