welcome to flyertalk!
when you go from one country to another, there is usually some sort of control point to decide who may enter the country. There are, of course exceptions.
One exception is often if you have a flight from country A to country B, stopping in country C. In many airports, you can remain in a holding area, that is considered "outside" country C for the purposes of accesses control, while you wait for your onward flight. I believe HND is one of those airports.
If you decide to leave the holding area, you'll have to officially "enter" Japan, which means going through the access control process. That's what people mean by "immigration". For US citizens, normally all that is required is a US passport, and filling out a form. If you decide to enter Japan, you should not take our word for it, and check this on your own so that you know the rules.
The main reason it can take a long time is if there are long lines. The actual time you spend interacting with an agent will most likely be very small. A few minutes at most. But, if a bunch of flights land at near the same time, lines can get long.
I think you should have time go to into the city and do something and get back in plenty of time for your flight, but I'll defer to a local expert on that. Transit costs will depend on where you go. I want to say that it should be $10-$15 for a round trip train ride to most of central tokyo. There are infinite things you could do, and the choice will depend heavily on your interests.