Thinking about this logically
If you want to go from Fukuoka to Hiroshima, the closer you are to Hakata Station, the better. If you were staying near Tenjin, you'd have to change trains at least once to get to Hiroshima and that change would happen at Hakata Station.
As for getting to the Nakasu red light district, I did it once over 10 years ago. I was staying at a business hotel very close to Hakata Station, and I was just passing through Fukuoka. It was in the evening and I decided to take a walk and look around, because it was my first time there. A good amount of people were out and about. It didn't take much walking before I walked over a bridge and found myself walking around Nakasu. I don't remember if I was looking for it on purpose or if I found it by accident. The main point I'm trying to make is that Nakasu was a short walk from Hakata Station.
As for Tenjin, I've never been in that area, but it does look nice. Also, looking at Fukuoka on a map, Tenjin to Nakasu looks about the same distance as Hakata to Nakasu. It might be walkable too.
As for the English situation, a lot can change in 10+ years, but back then it seemed like the further you were from Tokyo, the more Japanese you needed to speak. I don't know to what extent things have improved for non-Japanese speakers since then. | |