  Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a
distance $\ell$-proper path if no two edges of the same color can appear with
less than $\ell$ edges in between on $P$. The graph $G$ is called
$(k,\ell)$-proper connected if there is an edge-coloring such that every pair
of distinct vertices of $G$ are connected by $k$ pairwise internally
vertex-disjoint distance $\ell$-proper paths in $G$. The minimum number of
colors needed to make $G$ $(k,\ell)$-proper connected is called the
$(k,\ell)$-proper connection number of $G$ and denoted by $pc_{k,\ell}(G)$. In
this paper we first focus on the $(1,2)$-proper connection number of $G$
depending on some constraints of $\overline G$. Then, we characterize the
graphs of order $n$ with $(1,2)$-proper connection number $n-1$ or $n-2$. Using
this result, we investigate the Nordhaus-Gaddum-Type problem of $(1,2)$-proper
connection number and prove that $pc_{1,2}(G)+pc_{1,2}(\overline{G})\leq n+2$
for connected graphs $G$ and $\overline{G}$. The equality holds if and only if
$G$ or $\overline{G}$ is isomorphic to a double star.
