According to the Norwegian institute for air research, tomorrow is not a good day to go outdoors in the Kansai region or anywhere else in western Japan, unless you don’t mind taking a chance on getting irradiated.
According to them, radiation coming from the Fukushima plant will be passing over the Kansai region starting tonight and throughout the rest of the day tomorrow.
Click this site for the air pattern and map. Radiated Air-flow Map
Apparently the map is now unavailable, either because it was incorrect or causing a panic. Nonetheless, if any amount of radiation does flow over the Kansai region, it will most likely be so low an amount that it will not threaten health.
To dispel the panic that seems to be going on more outside Japan than inside, here is an article about the amount of radiation that a person is subjected to on a daily basis in Hong Kong compared to the amount people are being subjected to in Tokyo lately.