When I arrived at my closest train station, I found that an accident had been happeningoccurred on the train line.

March 15th - Public Transportation in Tokyo.

I'm going to talk about the public transportation in Tokyo.

Nothing is grammatically wrong with this sentence, but it's unnatural to start a piece of writing like this. You can put the topic in the title like I did, or you can just leave it out entirely and it will be fine.

Tokyo is one of the world's largest cities, and it has dense, ample,a robust and complex system of train lines.

Again, nothing technically wrong with what you wrote, but it sounds slightly unnatural.

They are very useful so that; you can reach almost anywhere in the city by train.

Using "so that" here implies that the train lines being useful is what causes the fact that you can go almost anywhere using the trains. It's more natural to say the other way around; you can reach almost anywhere in the city by train, so that's why they are very useful.
Use a semicolon here to link the two clauses without using connecting words.

However, there are proublems happen on a day-to-daon a daily basis. ¶
or,¶
However, troubles arise on a daily basis.

"Troubles happen" is understandable but not a very natural phrase.

When I arrived at my closthe nearest train station, I found that an accident had been happeninged on the train line.

We rarely use the past progressive tense to describe accidents. Usually an accident is something that happens in an instant, like getting in a car crash or dropping something fragile.
You can use the past progressive if you name a specific problem.
"The trains had been running behind"
"They had been having trouble with the engine"

I had to wait about an hour for it to resume working,. I spendingt the time drinking coffee in the cafe.

I advise you to check an alternative way to reach your destination when you come totraveling in Tokyo!