Metro Free Wifi
On the Tokyo Metro lines (Marunouchi, Hibiya, Fukutoshin, Hanzomon, Tozai, Yurakucho, Chiyoda, Nanboku), there is Metro Free Wifi.
On the Toei Subways, it is not available
Strong signal in station, weaker within tunnels
The service is OK if you happen to be outside the train, as the Wifi access points are generally located at the station themselves.
Sometimes, it is possible to use the Internet between stations, but this is largely dependent on how far the moving train is from the access point.
And as the train is moving, do not expect a stable signal.
If you need access to downloading biggish files, do it at a station. Don't even try when moving through the tunnels, the signal will break.
If you are trying to send big files, again the station.
If short e-mails, then you might have some luck when moving through the tunnels.