Question
o-nitrophenol is more acidic than $o-$methoxyphenol.

Answer

$-R$ and $-I$ effect of $-NO_2$ group decreases the electron density in $O-H$ bond and make loss of proton easy in $O-$nitrophenol whereas $+R$ effect of $-\ce{OCH_3}$ group increases the electron density in $O-H$ bond and makes release of proton difficult in $o-$methoxyphenol.
That is, why $o-$nitrophenol is stronger acid than $o-$methoxyphenol.

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