Depends on which Japanese you're talking about. And even then it's hard to say exactly how much they knew.

Familiarity with the current events of the US was not high on the priority list for most Japanese people. The average Japanese person would not know and would struggle to care if you told them about it.

They certainly knew that the American continents existed and that there was a polity named the United States. This is because of, as you say, the well known 'Dutch studies', ie information trade with the Dutch. Major events such as wars would be interesting news for merchants and rulers (merchants from anywhere have historically been keen to always stay up to date on major events that can affect trade, like wars). But distant wars that didn't affect one's home were likely no more than the subject of momentary curiosity and idle chatter.

The focus of these studies was prioritized immediate practically applied things, such as medicine, construction or agriculture techniques, military science, ship building, etc. Political, historical, and cultural studies were a distant second.

That doesn't mean that no one studied them at all, nor that the knowledge was totally absent, but due to the already limited amount of information that came by way of trading in Dutch articles (the Dutch had to know about it and then care enough to write about it) and the selective trade conducted by Tokugawa Japan (the Japanese had to care enough about it to trade for it), Japanese scholars generally would not come upon a great deal of information about the US.

When Perry's expedition arrived, the Japanese knew where they were from and who they were.

The Shogunate was at the very least aware of the US (what and where it was) by the mid-1840s. One of the figures I have been researching lately, Commodore James Biddle, traveled to Japan c. 1845 as a part of a broader diplomatic mission he had been charged with, focusing on China.

David Foster Long, who specializes in 19th century American naval officers and their use in diplomacy, refers to this as the most important and obvious antecedent to Perry's trip the following decade. His trip to the Far East was mostly focused on expanding American trade and diplomatic privileges with China, in the face of British gains recently won in the First Opium War. His instructions suggested he try, without exerting undue pressure or causing some sort of damaging diplomatic incident, to open formal diplomatic and commercial relations between the US and Japan (they knew from the start it was likely a long shot).

At the time, American vessels would occasionally arrive in Japan. Some attempting trade, some having maritime emergencies (such as shipwreck), and others wanting trade and merely feigning a maritime emergency ("Oh no! We're in Japan and have insufficient food and water to leave. Well, while we're here, want to trade?"). Occasionally some Americans would be captured and detained for making too poor a show of their feigned distress, which required American diplomatic intervention to get them freed.

When Biddle arrived in the mid-1840s (don't have the book in front of me, sorry) he stopped in a major harbor and was immediately told to leave, as normally occurred to unsolicited visitors. He responded by providing local political leaders with a copy of letter from the government of the United States, requesting Japan open up trade with us. As a matter of courtesy, local officials sent the letter to the Shogun and provided food and water (the latter being of such good quality that sailors rejoiced that the water was the best they had drunk during their entire trip, tasting neither foul nor stagnant).

The Japanese were courteous, though cold, while Biddle and his men waited for a response.

The response that arrived was fairly cold, though once again cordial, saying that Japan had no desire to open up trade since current policy had been in place for so long it was a matter of tradition (yes, that's more an excuse than a reason) and that the exception to this rule, the Dutch, was an interaction so limited that it didn't count (again, an excuse).

In the midst of this there was an incident where, due to mixed messages, Biddle attempted to disembark without authorization (he had been offered a chance to disembark, declined, then changed his mind and tried to do so without an escort) and was struck in the face by one of the Japanese soldiers in the numerous little launches that surrounded Biddle's main vessel. Biddle chose to not make an issue of the misunderstanding, for which the Japanese were grateful (it was Biddle's fault, in part, but striking a high ranking naval officer could have been turned into a diplomatic incident).

Biddle sailed away, having failed his mission.

This is a basic summary of the account related by David Foster Long in his biography of Biddle, titled Sailor-Diplomat.

What struck Long in his research of this specific incident was how ignorant Americans were of Japan. As /u/AsiaExpert pointed out, the Japanese roughly knew the who and the where of the US, but not much else. Even the Chinese during this period did not seem to care too much about what differentiated us from the British. We were a bit more interested, but still largely ignorant. In his research, Long noted that American correspondence focused on the figure of the Emperor, for example, despite the fact that the Shogunate had been the real power in Japan for well over a century. Letters, diaries, official diplomatic correspondence, all refer to the Emperor ruling Japan. To be fair, European literature on the Far East wasn't of the best quality at the time, and Japan was shut off from much of the rest of the world, so it isn't necessarily Biddle or the White House's fault that we were largely ignorant of the real power situation there.