Here are two more precise formulations of the problem which will allow for an approximation scheme as asked for by the OP.
Let $A \in \mathbb{K}^{N \times N}$ be a sparse matrix and well-conditioned matrix and $b \in \mathbb{K}^N$ a sparse vector. Then we can associate $A$ with a graph $G = (V,E)$ where
$$
V := \{1, \ldots, N\}, \qquad
E := \{(i,j) \in V^2 \mid A(i,j) \neq 0\},
$$
which in turns allows us to define a distance $d(i,j)$ between indices given by the minimal number of edges in $G$ we need to take in order to get from $j$ to $i$ (i.e., $d(i,j)$ is the graph distance in $G$). Then one can show
$$
|A^{-1}(i,j)| \leq C \, \exp\big(-\gamma \, d(i,j)\big)
\qquad \forall i,j \in V
$$
for some $\gamma > 0$, see e.g. "Decay Rates for Inverses of Band Matrices" by Demko, Moss and Smith (1984).
Given the above, it is clear that $x = A^{-1}b$ decays exponentially if we arrange its coefficients according to the graph distance from the non-zero entries of $b$, which is the ordering we assume for the following construction of an approximation $x_n = A_n^{-1} b$ with only $n \ll N$ degrees of freedom. Let $A_n$ be the matrix in which all interactions between entries $\leq n$ and $> n$ are dropped, i.e.
$$
A_n(i,j) := \begin{cases}
A(i,j) & \text{if } i,j \leq n \text{ or } i,j > n, \\
0 & \text{otherwise}.
\end{cases}
$$
It is clear that then $x_n(i) = 0$ for $i > n$, and hence it only remains to show that $x_n(i) \approx x(i)$ for $i \leq n$. This can be seen by expressing the error as
$$
x - x_n = (A^{-1} - A_n^{-1})\, b = A^{-1} \, (A_n - A) \, A_n^{-1} \, b.
$$
We now assume that $A_n^{-1}$ satisfies the same decay estimate as $A^{-1}$, which boils down to assuming that the conditioning of $A_n$ is not worse than that of $A$. This is trivial if $A$ is positive definite but may be hard to verify otherwise, yet it is an assumption you would probably want to make anyways if you want to numerically solve $A_n x_n = b$. Under this assumption, we estimate the error as
$$
|x(i) - x_n(i)|
\leq
C \, \sum_{k > n} \sum_{\ell \leq n} \sum_{j \in \mathrm{supp}(b)} A(k,\ell) \, \exp\big( - \gamma (d(i,k) + d(\ell,j)) \big),
$$
which is at most $C \, Nn|\mathrm{supp}(b)| \, \exp(-\gamma \, d(n+1,\mathrm{supp}(b))\big)$ and can be as small as $C \, Nn|\mathrm{supp}(b)| \, \exp(-2\, \gamma \, d(n+1,\mathrm{supp}(b))\big)$ for $i$ close to $1$.
Alternatively, if $A$ is not sparse but $x$ is known to be decaying for other reasons, we can still order the degrees of freedom such that $|x(i)|$ is decreasing for increasing $i$. We then define the projector onto the first $n$ degrees of freedom,
$$
P_n : \mathbb{K}^N \to \mathbb{K}^N,
\qquad
a \mapsto \begin{cases}
a(i) & \text{if } i \leq n,\\
0 & \text{otherwise},
\end{cases}
$$
and set $A_n := P_n A P_n$ and $x_n := A_n^\dagger b$, where $A_n^\dagger$ is the pseudo-inverse of $A_n$ satisfying $A_n^\dagger A_n = A_n A_n^\dagger = P_n$. Note that once again $x_n$ is formally an element of $\mathbb{K}^N$ but has only $n$ non-zeros. We can then write the error as $x_n - x = (x_n - P_n x) + (P_n x - x)$. The term $P_n x - x$ is easily estimated using the decay of $x$, and for the other term we get
\begin{aligned}
x_n - P_n x
&=
P_n \, (A_n^\dagger - A^{-1}) \, b
\\&=
A_n^\dagger \, (A - A_n) \, A^{-1} \, b
\\&=
A_n^\dagger \, (A - A_n) \, x
\\&=
A_n^\dagger \, A \, (x - P_n x)
\end{aligned}
and can hence be estimated as well by the projection error $x - P_n x$ if $A_n^\dagger$ and $A$ are bounded in suitable norms.
