The rule for choosing between a and an depends on the sound that begins the next word, not the letter itself. The word anthropologist begins with a vowel sound, the short a sound, so it must be preceded by an, not a. This gives us An anthropologist by profession, he is also a trained classical singer.
Option 1, A anthropologist, is wrong because a is only used before words that start with a consonant sound. Option 2, The anthropologist, changes the meaning by making it definite and specific, as if there is only one particular anthropologist already known to the listener, which does not fit an introductory descriptive sentence like this one. Option 3, As anthropologist, is grammatically broken because as needs to be followed by a full noun phrase such as as an anthropologist, not a bare noun.
Only option 4 follows the correct article rule and reads as a grammatically complete sentence, so it is the right choice.