This is a great question and reasonable to ask if you’ve watched the musical. As the musical shows, that by the time of Hamilton's death, he underwent both a serious scandal following his own publication of the 95-page Reynolds Pamphlet and his multiple public confrontations between other leading figures, like Adams and Jefferson. However, Hamilton was still a very-well known figure and his death was seen as a huge shock to not only New Yorkers, but many Americans spread throughout the United States.
By his death in 1804, Alexander Hamilton was a very well-known figure across America. First made famous for his participation in the victory at Yorktown in 1781, Hamilton rose through the public ranks quickly through the 1780s and 1790s. His political and legal career also kept him constantly in the spotlight, both by friends and enemies typically through the press. People knew him not only for these accolades, but also as someone who was profoundly influential in drafting the new US Constitution and also was well-respected by George Washington, whom everyone still adored. So the shock and sadness, as described in the musical wasn't only shared by people who supported Hamilton, but a general sense of horror that such a well-known figure could be killed in a duel.
As Ron Chernow explains in his biography of Hamilton, "When a handwritten notice of Hamilton’s death went up at the Tontine Coffee House, the city was transfixed with horror... Even Burr’s friend Charles Biddle conceded that “there was as much or more lamentation as when General Washington died... Unlike at Washington’s death, however, the sorrow was laced with shock and chagrin at the senselessness of Hamilton’s demise."(1)
Hamilton's death gripped Americans everywhere. Newspapers ran headlines going over the unexpected death and church bells rung out in cities like Philadelphia and Boston to mourn his passing. As Chernow, and other biographers explained, many factors led to this outpouring of support. Among them, Hamilton was only 47* when he died, he left behind a young grieving family with many several children, his history of service to America echoed in people's minds and his painful end at the hands of the Vice President enabled Hamilton to finally, "achieve in death what had so often eluded him in life: an emotional outpouring of sympathy from all strata of New York society."(2)
Biographers describe New York in July, 1804 as being in grieving as a city because they lost their 'most distinguished citizen.' The New York Common Council urged all businesses to be closed on the day of his funeral, which turned into a huge spectacle. Not only was the crowds huge for the procession, but New York militia came out to bear arms and do military drum rolls and ships in the harbor few their sails at half-mast. Chernow explained, "It was the grandest and most solemn funeral in the city’s history to date." (3)
Ultimately, the musical didn't exaggerate that New Yorkers and Americans in general were shocked at Hamilton's unexpected death and openly grieved together.
*Hamilton's date of birth is in dispute. Some sources believe he was born in 1755 and others in 1757, which means he was either 47 or 49 when he was killed.
Chernow, Ron. Alexander Hamilton Penguin Publishing Group. pp 711
Ibid
pp 712
edit: added a link
I haven't seen the play, so I can't comment on it directly, but, yes, there was an outpouring of public grief upon the death of Alexander Hamilton, particularly in his adopted hometown of New York.
Hamilton is the founder of the New York Post (New York Evening Post at the time), and on the evening of June 13, 1804, the paper published a letter from the Rev. Benjamin Moore, the clergymen who attended Hamilton as he died. Moore began his letter by acknowledging that "the public mind" was "extremely agitated by the melancholy fate of that great man, Alexander Hamilton".
The following day, William Coleman, the editor of the Post and good friend of Hamilton's, couldn't really add much more, writing: "As soon as our feelings will permit, we shall deem it a duty to present a sketch of the character of our ever-to-be-lamented patron and best friend". Instead of writing anything new, they just reprinted Moore's letter again.
Alexander Hamilton's father-in-law, who was bedridden in Albany with the gout, wrote to his newly-widowed daughter immediately, lamenting that he could not get down to New York due to his condition, but as soon as he was better he would. He instead invited her to come to Albany: "I entreat you my beloved Child to come home as soon as you possibly can, with my dear Grand-children."
James Cheetham, editor of the New York American Citizen, the Republican paper who had played a central role in goading the two men into the duel, was quick to distance themselves from what happened:
"Wrap[ped] up in himself—to appease his resentment, to gratify his ambition, [Burr] is capable of wading through the blood of his fellow citizens and of laughing at the lamentations of widows and orphans."
The newspaper called Burr's actions "predetermined hostility". The New York Evening Post would say Hamilton was "willfully and maliciously MURDERED by the hand of AARON BURR" in an editorial a couple weeks later.
Ron Chernow's biography, Alexander Hamilton, probably gives the best account of the aftermath:
"When a handwritten notice of Hamilton’s death went up at the Tontine Coffee House [on Wall Street], the city was transfixed with horror. 'The feelings of the whole community are agonized beyond description,' Oliver Wolcott, Jr. [Hamilton's successor as U.S. Secretary of the Treasury], told his wife. New Yorkers of the era never forgot the extravagant spectacle of sadness, the pervasive grief. Even Burr’s friend Charles Biddle conceded that 'there was as much or more lamentation as when General Washington died.' As with Washington, this mass communal sorrow provoked reflections on the American Revolution, the Constitutional Convention, and the founding of the government. Unlike at Washington’s death, however, the sorrow was laced with shock and chagrin at the senselessness of Hamilton’s demise."
Chernow writes that for the rest of its term that year, "the New York Supreme Court draped its bench in black fabric, while the Bank of New York building was also sheathed in black. For thirty days, New Yorkers wore black bands on their arms."
Nathan Schachner's biography Alexander Hamilton gives a similar account:
"The city went into mourning, and the nation. The newspapers out-rivaled each other in expressions of sorrow. The clergy preached long sermons, with the duel as their text. Mass meetings were held, in New York, in Philadelphia, in Boston, in Albany. Church bells tolled...
"In New York City the merchants and citizens gathered at the Tontine Coffee House to mourn their loss. On the date of the funeral they closed all stores and marched in vast procession to the muffled beat of drums. The ships in the harbor half-masted their flags. The City Council attended in a body, and the members of the bar, party lines forgotten, decreed mourning for a period of six weeks. The mayor, members of congress, foreign ministers, the students of Columbia, the Cincinnati, Tammany and all the citizenry marched in line."
Rev. Benjamin Moore led the service, which was held at Trinity Church on Broadway. U.S. Senator Gouverneur Morris, and fellow delegate to the U.S. Constitutional Convention, gave the eulogy and it was observed he was so broken up, the attendants weren't sure if he would be able to get through it.
In the second volume (Alexander Hamilton: The National Adventure, 1788-1804) of his lengthy two-volume biography, author Broadus Mitchell's account is largely the same as Chernow's and Schachner's. He writes that Aaron Burr spent eleven days housebound in his New York City home after the news broke, eventually fleeing with the help of some friends in the dark of night because "it was feared that a mob would burn" his house down. (Note: Chernow says it was only nine days. Burr snuck out two days before he fled to have a sexual liason.)
Mitchell writes that there was an outpouring of grief and memorializing in newspapers around the country. He quotes from Fisher Ames' eulogy in the Boston Repository as being one of the most "thoughtful" and "perceptive":
"[Hamilton] had not made himself dear to the passions of the multitude by condescending ... to become their instrument. ... it was by ... loving his country better than himself, preferring its interest to its favor, and serving it, when it was unwilling and unthankful, in a manner that nobody else could, that he rose, and the true popularity, the homage that is paid to virtue, followed him."
But Mitchell does explain that the memorials weren't universal, led by the pro-Jeffersonian New York Chronicle:
"Of course, Burr found apologists too. The N.Y. Chronicle declared that when the public was candidly informed, Burr would 'be justified by every disinterested ... man'. A Boston paper objected to a eulogy of Hamilton: 'An Oration! The Champion, the Goliath of party is dead and died like a fool! He ought to have the burial of an ass, and none to lament him...'"
So, while there were some newspapers that stuck to their anti-Hamilton guns, so to speak, that kind of sentiment was only expressed in a small minority of newspapers.
In summary, yes, if the play shows this kind of outpouring, then they got it right. Considering the play is a loose adaptation of Chernow's book, they picked a good source. Some of the briefer biographies of Alexander Hamilton end abruptly with the circumstances of his death, with an epilogue of what happened to his wife, and to Aaron Burr. They don't say anything about the public reaction at all.
SOURCES:
Chernow, Ron. Alexander Hamilton, 2004.
Cray, Ed., ed., et. al. American Datelines: Major News Stories from Colonial Times to the Present, 2003.
Mitchell, Broadus, Alexander Hamilton: The National Adventure, 1788-1804, 1962.
Nolan, Charles J. Aaron Burr and the American Literary Imagination, 1980.
Schachner, Nathan. Alexander Hamilton, 1946.