According to Molecular Orbital Theory, bond length is inversely proportional to bond order; as the bond order increases, the bond length decreases . To find the increasing order of bond length, we must determine the bond order (B.O.) for each species:

1. $NO^{+}$: This species has 14 electrons and is isoelectronic with $N_{2}$. Its bond order is 3.0 .
2. $NO$: This molecule has 15 electrons. Compared to $N_{2}$, the additional electron enters an antibonding $\pi^{*}$ orbital, reducing the bond order to 2.5 .
3. $NO^{-}$: This ion has 16 electrons and is isoelectronic with $O_{2}$. Following the molecular orbital configuration of $O_{2}$, it has a bond order of 2.0 .
4. $O_{2}^{-}$ (superoxide ion): This species has 17 electrons. The extra electron further occupies antibonding $\pi^{*}$ orbitals, resulting in a bond order of 1.5 .

Comparing the bond orders: $NO^{+} (3.0) > NO (2.5) > NO^{-} (2.0) > O_{2}^{-} (1.5)$. Therefore, the increasing order of bond length (the reverse of bond order) is $NO^{+} < NO < NO^{-} < O_{2}^{-}$.