I've been wondering about the "warning time" thing, too .... both with the earthquake itself and the tsunami.
Japan is fortunate to have one of the best geophysical monitoring systems on earth. In a case like this, where the earthquake's hypocenter is several hundred km away from major cities like Tokyo, they can actually detect the fast-moving P waves a minute or more before the damaging but slower S-waves arrive. They take advantage of this lead-time to slow down trains, institute emergency procedures in industrial facilities, etc. Of course, if there were a similar large quake closer to Tokyo, the difference in arrival time between the P-waves and S-waves might only be a few seconds, in which case the earthquake warning system would be much less effective.
The tsunami warning is a bit more complicated. The first warning in Japan was about 3 minutes after the quake. In Hawaii, NOAA took a few minutes longer because they do modeling of the tsunami's propagation over the entire Pacific basin -- their first warning was 9 minutes after the quake.
It's strangely difficult to obtain actual hard information about the timing of the tsunami's arrival at specific points along the coast. The initial NOAA forecast predicted arrival at 15-30 minutes after the quake. At Sendai airport, it actually was over an hour before the tsunami broke over the shore berm and began flooding the airport.
JMA has a table of arrival times for the "initial tsunami" and "maximum tsunami" here
, though it's formatted in a not particularly legible way.
There are a bunch of places on the north coast that reported an "initial tsunami" within 10 minutes after the quake ... but these first waves were generally 0.3 m or less ... except for some of the hardest hit places, where the initial amplitude is "unknown", probably because the recording equipment was wiped out soon afterwards.
There's then a pause until 29 minutes after the quake, when other places begin reporting their initial tsunami observations ... and Ofunato reported its maximum height of "3.2 m or more". Many others don't report the maximum wave arrival until more than an hour after the quake (like Sendai airport).
Basically, there seems to be a lot of variation in the arrival times. I think that's because the shape and period of the wave interact with the bathymetry and shape of the coastline to amplify the wave in some areas and dampen it in others.
Bottom line, a few communities might have had little warning, but most had at least 25 minutes, and many places had an hour or more to evacuate. At least, that's what it looks like to me.