Hi,
Yes, it can be done although it is a pretty busy schedule. A 7 day pass for Days 3-9 would save you a ton of money.
A few observations on your plans though. The Ashikaga Flower Park is an absolute gem of a place and beautiful regardless, but if you want to see the wisteria, generally it is in bloom in late April through the first 10 days of May. Golden Week is a very busy holiday period during that time as well and it can get crowded at many tourist sights.
For Day 5 the train is cute but if you have not seen Kyoto or Nara, they are far more interesting and the cultural jewels of the country - worth as much time as you can spare. Kinkakuji is gorgeous, the Fushimi Inari Shrine is unforgettable, and the Todaiji Temple will leave you awestruck.
For your Day 6 it is possible to see the best sights of Hiroshima City if you start early (the Peace Park/Museum, castle, Shukkeien Garden, and shopping arcade+okonomiyaki mura) and then spend the bulk of the day to see Miyajima, which is breathtaking. In the evening you can continue to Fukuoka and try the tonkotsu ramen along the Naka River.
For Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. See this site as a good guide.
Don't miss the Nanzoin Temple which is one of Japan's finest yet almost totally unknown to the outside world and completely free!
Look into a Fukuoka Tourist City Pass also.
You can also see a lot of previews on the best there is to see here.
For Day 8, if you care to see it, Kobe and Himeji make an excellent day trip. The latter has Japan's best castle. For Day 11, you might consider a day trip to Nikko, Kamakura, Hakone, or the Fuji 5 Lakes area. There are regional rail passes for all of them.
To get to/from the airport cheaply, look into a 1000 yen bus.
Best of luck!