Approximation theorem for Anti-Self-Dual Metrics Rounge's Theorem states that any meromorphic function on a domain inside $\mathbb{C}$ can be approximated (over compact subsets) by a sequence of rational functions (meromorphic functions on $\mathbb{C}P^1$).  
Donaldson wrote a small paper "Approximation of Instantons" based off of Taubes' gluing construction for instantons on $S^4$, which gives an analog to Runge's approximation theorem for meromorphic functions. There the statement is that given an open subset $U\subset S^4$ and an instanton $A_U$ over some principal bundle $P_U\to U$, there is a sequence of bundles $P_n\to S^4$ and instantons $A_n$ (and bundle maps $P_U\to P_n$ restricted to $U$) such that the pullbacks of $A_n$ converge to $A_U$ (over a slightly smaller subset of $U$).
Now Taubes has a similar construction to build anti-self-dual metrics: Given a compact oriented 4-manifold $X$, there exists ASD metrics on $X\#_n\mathbb{C}P^2$ (meaning connect sum with $n$ copies of $\mathbb{C}P^2$) for $n>>0$. This motivates:
In a similar spirit, is there likely to be a (Runge-type) approximation theorem in the realm of ASD metrics on 4-manifolds? I want to propose:
Let $X$ be a compact oriented 4-manifold. Given an open subset $U\subset X$ and an ASD metric $g$ on $U$, there is an integer $R$ and a sequence of ASD metrics $g_n$ on $X_n=X\#_{R+n}\mathbb{C}P^2$ and inclusions $U\to X_n$ such that the pullbacks of $g_n$ converge to $g$.
My immediate concerns/thoughts:
1) For this statement to even make sense I wouldn't be able to connect sum $\mathbb{C}P^2$ into the region $U$.
2) If Taubes' construction can build an ASD metric on $X_n$ which doesn't touch $g$ on $U$, then my desired statement is true.
 A: I think that your point 2) is more or less correct. In his proof Taubes first constructs approximately ASD metrics $g_N$ on $X \#_N CP^2$, starting with a metric $g$ on $X$, and this step would not touch the metric on $U$ if $g$ is already ASD there. These will be metrics with 
$$ \lim_{N\to \infty} \Vert W^+(g_N)\Vert = 0 $$
in a suitable norm, and $g_N = g$ on $U$, i.e. no connect sum is performed in $U$. An extra layer of connect sums is performed to ensure invertibility of the linearized problem (as opposed to just decreasing $W^+$), but I suspect that one can do this without touching $U$ as well. 
Then the perturbation to a genuine ASD metric is global, and once $N$ is large enough, you should end up with ASD metrics $g_{N,ASD}$ on $X\#_N CP^2$, such that
$$ \lim_{N\to \infty} \Vert g_{N,ASD} - g_N \Vert =0 $$
in a suitable norm. The choice of this norm is such that $g_{N,ASD}$ will in particular be very close to $g_N = g$ over $U$ in the $C^0$ sense once $N$ is large. Since both $g_{N,ASD}$ and $g$ are ASD metrics over $U$, one should be able to get improved regularity there and conclude that on any compact subset $K\subset U$ the metrics $g_{N,ASD}$ converge to $g$ in any $C^k$ norm.
So in sum I believe that the answer to your question is yes, but of course one would need to go through Taubes' argument to make sure. 
A: There are some results in this general direction, but on non-compact manifolds.  See for instance Instanton approximation, periodic ASD connections, and mean dimension, by Shinichiroh Matsuo, Masaki Tsukamoto and some papers they reference in there.
