My sister has just got . . . . . . . . job at . . . . . . . . bank in Manchester.

'A' is used before unspecific/general nouns that begin with a consonant sound. Here, 'job' and 'bank' are unspecific ones. Hence, option C is correct.
Had there been only one particular bank in Manchester, we would have used 'the' in the second blank. However, it is unlikely that there would be only one bank.