Jamie would not know the word "hangover" as it related to the after effects of alcohol consumption. This sense of the word dates to the 1890s and is of American origin. It is, however, possible he could infer the meaning by context.
Claire tells Jamie she can't marry him because she doesn't know his real name. In the previous episode "The Garrison Commander" Claire reads over the "contract of marriage" where both her name and Jamie's full name is listed.