Do you capitalize drama as a subject in school?

No. Drama begins with a 'D' at the beginning of the sentence, the rule 'first Word of a sentence is capitalised' applies. But drama in an ordinary sentence is an ordinary noun of a subject, and therefore not capitalised. The fact that it is a 'name' of subject does not make it a proper noun (like John, or Arianna, or England). The school subject of English is already a proper noun as it refers to the language of England and follows the capitalisation of that proper noun, as do other derivations like Englander. [A proper noun is the name of one specific person or country, but not an abstract entity and properties like justice or energy.) It is easy to get confused with the practice of using title case in typsetting of titles, eg 'English and Mathematics'. This is a matter of house-style, a text book or the Times newspaper may not follow it, a magazine might. Once a house-style is selected, the editor will enforce it. The choice of the house-style is subjective and only overrides the normal rules of grammar and accidence on presentational grounds. For professional standards on proofreading/house-style/etc consult something like 'Hart's Rules for Readers and Compositors at the Oxford University Press'.