Hi,
You could do it either way - but there are a few possibilities here if you are interested. You are generally going to Japan too early for the cherry blossom season. But you might still catch some good blooms from the plum blossoms (ume) that precede it. There are some very good places to see them, and late February to early March is often the best time. Kairakuen in Ibaraki, just north of Tokyo, is one of Japan's Top 3 traditional historic gardens, with about 100 different types of ume and about 3000 trees - it is one of Japan's very finest. If you are interested, that might make flying into Tokyo more attractive. There are others as well, including the Koishikawa Korakuen garden, Hanegi Park and Yushima Tenjin Shrine, all in Tokyo.
In the Kansai area, there is the Kitano Tenmangu Shrine in Kyoto with about 2000 trees. And since you mention Fukuoka, there is another nationally renowned place in nearby Dazaifu.
It could warm up a bit during your time in Japan, but there is not a large temperature variation between Kansai and Tokyo. That said, the flowers follow their own schedule, and if you get lucky - maybe, just maybe, could catch the start of the cherry blossom season. It is impossible to predict the starting time with certainty until about 4-6 weeks before the blooms first start to come out. If you want to chance it, the places to be at the end of your trip would be as far south as possible (such as Kyushu), or for your itinerary, in Tokyo - which paradoxically starts its cherry blossom season a couple of days on average before Kyoto which is farther south.
If you can go to Fukuoka, it is well worth it. You'll find many open parks, shopping, seaside historical areas, and urban neon. Look at this site as a good guide. Don't miss the Nanzoin Temple which is one of Japan's finest yet almost totally unknown. The Rakusuien and Yusentei Gardens are both beautiful. Nagasaki though is really the most interesting city on the island. It is worth a couple days to see it.
It is understandable to avoid Osaka during the day, since you are already seeing Tokyo which is more interesting. But at night the city comes alive and has some great places to see, such as Dotonbori, plus the night views from the Umeda Sky Bldg, and Abeno Harukas Bldg are terrific. Don't skip Nara either - the Todaiji Great Buddha is a must-see place, and feeding the hungry deer in Nara park is something you won't forget.
Best of luck.