Take $Q_n;=\{q_1,q_2,...,q_n\}$ the topological space with n points, paired with the discrete topology.
I need to find a 2-fold covering of $p:Q_4\rightarrow{Q_2}$.
If I defined $Q_4=\{x_1,...,x_4\}$ and $Q_2:=\{y_1,y_2\}$, would the function $p(x_i):=\{y_1$ if $i\in\{1,2\},y_2$ if $i\in\{3,4\}\}$ work as a two fold covering?
I don't clearly understand the requirement that $p^{-1}(U)$, with $U$ some open set in cover of $Q_2$, is a disjoint union of open sets in the cover for $Q_4$.
In fact each surjective map $p : D' \to D$ between discrete spaces is a covering map. To see this, note that each $d \in D$ has $\{d \}$ as an open neighorhood and $p^{-1}(\{d \}) = \bigcup_{d' \in p^{-1}(\{d \})} \{d'\}$ where the $\{d'\}$ are pairwise disjoint open subsets of $D'$ which are mapped by $p$ homeomorphically onto $\{d \}$.
In your example you have a surjection whose fibers have $2$ elements, thus it is a twofold covering map.