Transformation of acyl group to carboxyl group in biphenyl system with two acyl groups at para position
Is it correct that you are aiming to synthesize 4-(4-hydroxyphenyl)benzoic acid [58574-03-1]?



The target compound is available in one step from biphenyl using potassium peroxydisulfate $(\ce{K2S2O8})$ as an oxidant, Palladium(II)-trifluoroacetate, $\ce{Pd(OCOCF3)2}$, as a catalyst, and formic acid as the carbon source for the $\ce{-COOH}$ group. The synthesis was published in
 Org. Lett., 2004, 6, 2437-2439 (DOI).



EDIT

Limited resources and experience will probably rule out Stille-type coupling reactions too. 

Alternatives pretty much depend on which 4,4'-disubstituted biphenyl are available.

A methoxyarene can be converted to the corresponding phenol by $\ce{BBr3}$
or the cheaper hydrobromic acid. The $\ce{-OCH3}$ group resists oxidation, is a +M substituent and is not acidic. All this might be helpful (see below).

In order to introduce the $\ce{-COOH}$ substituent, two approaches seem to be possible in an undergrade lab:


oxidation of a methyl group by permanganate  
Grignard reaction of a bromoarene with carbon dioxide