I answered this when they question was still a Japanese missile boat. But the editing shows what I consider the main misconception underlying the question.
Think People, Knowledge, Politics
Assume that the time travelers decide to support a local government and that they can convince the locals to trust them and to take them seriously to start with. For the latter, a warship will help, I'll admit that. Then the best the time travelers can do to support a local faction is to teach them technology just a bit ahead of everyone else. "Give a man a fish and you feed him for a day; teach a man to fish and you feed him for a lifetime."
A graduate student with a solar-powered notebook and the right collection of textbooks and blueprints should be more efficient in this regard than the books which happen to be on a small to medium warship. A large one would probably have a nice library and a good group of engineering officers. The key is to bring improvements which can be applied to a local tech base. A very good javascript programmer would have learned the wrong things for programming ENIAC. But blueprints for a CODAG power plant would make a real difference in the 1950s, and the formula for prismatic powder would be great in the 1850s.
The tech difference for "this makes a great prototype" is smaller than the tech difference for "this warship is a war winner."
Expanding on the answers by Thorne and DWKraus:
- 21 well-trained crew. It would be extremely silly to send them out using their ship, and risking their life. Send them to universities where scientists and engineers can pick their brains. The officers and petty officers won't know enough to manufacture new radars or gas turbines, but they can give pointers in the right direction.
- On the other hand, what makes you think that present-day Japanese would bow to the military dictatorship in the name of the Emperor and not support the forces of Democracy? Drop the boat into the battle of Midway and it might well support the USN after some serious soul-searching. Or stay out of it and try to reach Sweden.
- In addition to the crew, there would be reference books. Technical, historical, and so on. The butterfly effect might soon invalidate history books, however.