1. Identify the reaction type

   Benzene diazonium chloride undergoes azo coupling with activated aromatic compounds like phenols and naphthols.

   Here, the coupling partner is 1-naphthol in alkaline medium.

2. Effect of alkaline medium

   In alkaline medium, $1$-naphthol forms the naphthoxide ion, which is even more reactive toward electrophilic substitution:

   $\text{1-naphthol} \xrightarrow[\text{alkaline medium}]{} \text{1-naphthoxide ion}$

   The diazonium ion, $\text{C}_6\text{H}_5\text{N}_2^+$, acts as the electrophile.

3. Position of coupling in 1-naphthol

   In $1$-naphthol, azo coupling occurs predominantly at the 2-position (the position adjacent to the $-OH$ group), because this position is highly activated.

   So the product formed is:

   $\text{benzene diazonium chloride} + \text{1-naphthol} \rightarrow \text{2-(phenylazo)-1-naphthol}$

4. Nature of the product

   The structure is an azo dye having the linkage:

   $\text{C}_6\text{H}_5-N=N-\text{naphthol ring}$

   with the $-N=N-$ group at the 2-position of $1$-naphthol.

5. Match with the correct option

   The option representing 2-(phenylazo)-1-naphthol is Option C.

Final Answer

The coupling product is 2-(phenylazo)-1-naphthol, so the correct option is C.