Nucleophilicity in polar aprotic solvent (DMSO)
let me start with nucleophilicity of halides. In protic polar solvent, $\mathrm{F^-}$ is more solvated (hydrogen bonding) than $\mathrm{I^-}$ and hence less reactive. 

In polar aprotic solvent $\mathrm{F^-}$ solvation is less and therefore more reactive.

Such behavior may not be applicable here, since phenoxide is resonance stabilized while other compound is not. Therefore phenoxide $\ce{2}$  is less nucleophilic then cyclohexanolate $\ce{1}$.