He is . . . . . . . . honourable man in the locality.

'A' and 'An' are indefinite articles and 'the' is the only definite article in the English language. 'The' is used with reference to something specific such as a specific person or place. This sentence does not talk about a specific honourable man, hence 'the' will not be used.
'A' cannot be used with honourable because it begins with a vowel sound. By rules of grammar, 'an' is used before a word that begin with vowels or vowel sounds. The word honourable begins with a vowel sound (o) because H is silent, similar to the pronunciation of 'hour.' Hence, Option B 'an' is correct.