In ortho nitro phenol, electron withdrawing nitro group is present in ortho position, the electron density of O−H bond is reduced.
This makes removal of proton easy. After deprotonation, o-nitrophenoxide ion is resonance stabilized. Hence, ortho nitrophenol is a stronger acid.
In ortho methoxyphenol, electron releasing methoxy group is present in ortho position, the electron density of O−H bond is increased.
This makes removal of proton difficult. Hence, ortho methoxyphenol is less acidic.