Assuming that you read an average amount of newspapers and kept up with the political gossip of the time, a fair amount. Not many newspapers from 1805 survive in New York (nor anywhere else); not nearly all newspaper archives in New York are digitized, and of that set a smaller subset is digitized with OCR, but a simple search of the New York State Newspaper archive turns up 30 mentions of "Napoleon" in 1805, and 130 of "Bonaparte." Even taking into account that the archive won't return alternate spellings ("Buonaparte") or return words that are hyphenated (e.g. during line breaks), you see quite a few mentions of Napoleon starting in late October and November 1805, as news of the battle of Ulm reached the Americas and was distributed through newspapers.

Not every story by a long shot in those surviving newspapers was about Napoleon, but events on the Continent got a lot of play in American newspapers at the time, and they would pick up and reprint correspondence from other newspapers (copyright as we now know it was not A Thing then, nor were bylines for the most part).