Your question is an extremely short yes/no question, so allow me to answer this briefly, as it is difficult to provide a more complicated answer. Emphatically, yes, you do know what is going on in Europe on about a two month time delay, if you cared enough to read the newspaper offered for sale on the corner. The information was certainly available, especially to someone in New York City, and the minutiae of the war in Europe would be published as soon as it received.
In fact, you could be a citizen of, say, Cazenovia, N.Y., and receive updates about the progress of the war in Europe as well. In general, you'd have a more instantaneous piece of news, such as a report from a ship captain, with a much longer follow-up article sometime later which usually consisted of reprints of speeches, formal declarations, etc.
For example, two articles appeared in "The Pilot" of Cazenovia, New York on June 28th, 1809:
The first article was a reprint of a formal speech given by a French general (Davout probably, but doesn't have the modern spelling in the newspaper) to allied soldiers in the Confederation of the Rhine. The information about this declaration of war was probably already published earlier, but here they are providing much more detailed information as a follow-up to their readers:
"FOREIGN INTELLIGENCE
Proclamation by DAVOUSF - To the army of the Rhine. Hemau, April 12th.
Soldiers, his Majesty, the Emperor of Austria commenced hostilities on the 8th ----. This was announced by his General in Chief on the 9th. They have taken up arms - The orders of the day delivered to the Austrian armies are merely the effusions of futility. This is not the way to attack the soldiers of the Emperor Napoleon. They threatened to overwhelm us with defeat and disgrace upon the plains of Ulm and Marengo. Your conduct will show what right they have to make use of these threats. Soldiers, our beloved Sovereign, when he was first consul, offered peace - the Emperor of Austria refused it - Marengo compelled him to accept of terms. The English broke the treaty of Amiens. Our Sovereign had collected his armies on the French cost and the Emperor of Austria availed himself of that juncture, and without any previous declaration of war, violated the territory of our allies and threatened ours. Ulm compelled the enemy of our Sovereign o again make peace. In the present instance, the Emperor of Austria has threated the territory of our allies, without the least appearance of any difference between the two powers, and conceiving himself in a situation to undertake hostilities, has actually commenced war.;
...."
The second article provides another kind of update, an apparent reprint from some information from London, which reports on some hearsay and speculation of a big battle.
"London, May 2nd
Two vessels arrived at Yarmouth yesterday morning from Holland, the captains of which state that Bonaparte was endeavoring to enforce the conscription in Holland, that in many places the people had manifested a disposition to resist it by force.
The firing and illumination on the French coast on Friday night and Saturday morning has created apprehensions that the Austrians have been defeated. It is hardly possible that any decisive battle should have been fought so soon after the Austrians had cross the Inn, which was on the 10th of last month.
Some engagements between advanced p----, however, may have taken place, and there can be no doubt that Bonaparte would, in his present critical situation, magnify to the utmost partial advantage which he might gain."
The original source is here: https://nyshistoricnewspapers.org/lccn/sn83031475/1809-06-28/ed-1/seq-3/
If we want to look at New York City papers specifically and take randomized examples, you can usually find that they would report on rumors from the waterfront. For example, let's look at the July 20th, 1809 edition of the New York Evening Post. Napoleon entered Vienna on 13 May, and this news is only just beginning to reach New York, but it is reported on extensively in this issue.
"Arrived this morning, schooner Ann Kelly, from Halifax, bringing London dates to May 25th.
....
Hague, May 22nd,
On the 15th of April, the Emperor of Russia declared war against Austria and ordered his armies to enter Galicia and Hungary. The Russians have already entered Gallicia.
Since the 9th l --- the French flag has been unfurled on top of St. Stephens church at Vienna, on the 10th or 11th, the Emperor Napoleon was to leave the capital. The Emperor Francis II has solicited peace.
Dutch papers of the 22nd arrived late last night and have brought an account of the French having entered Vienna.
,,,"
Source: https://nyshistoricnewspapers.org/lccn/sn83030385/1809-07-20/ed-1/seq-2/
If you fast forward, you can also see that the newspapers would sometimes print false information about the progress of the War in Europe as well, although this one was well qualified in advance.
New York Evening Post of July 14th, 1810:
"The following letter was received at this office this morning, with the Baltimore postmark on it and charged 17 cent. Whether some one has attempted to play us such a trick as was played on Mr. Relf some days since relative to the negro conspiracy, or whether there has actually been an arrival at Baltimore, as mentioned, time must determine. The news cannot be true unless there is some mistake in dates, as we have had later news from Spain and Portugal than the 7th of May. We understand a simlar letter was received this morning at the office of the Mercantile Advertiser from Baltimore.
'Port of Baltimore, July 12th. Arrived brig Rockingham McClean, 74 days from Liverpool, dry goods and glass - Capt. McClean reports the following important news - May 7th, Lat 48, Long 16, was spoken by the British Frigate Ethalion, Captain Cochran, with a number of transports, having on board 8 or 10 thousand French prisoners, among whom was King Joseph, who was surrounded and taken at St. Marys - Captain McClean says he was ordered on board the frigate and saw the king, who he knew, having seen him before."
Source: https://nyshistoricnewspapers.org/lccn/sn83030385/1810-07-14/ed-1/seq-2/
You can also easily access many other historical newspapers such as this from the NYS Historic newspapers collection and feel what it was like to be a citizen at the time by flipping through the daily news.