In this case bout means boxing match

“her gold medal bout” is a phrase. Try putting a comma after it, to help you read the sentence.

Around and about are similar, in that round and bout are both nouns. Bout comes something meaning bend, specifically from a circuit of a plow making a field. A bout then takes on the exact same meaning as a round etymologically, though they mean different things in boxing. https://www.etymonline.com/word/bout