I think you're out of luck. Reports from others is that the 1.6.2 upgrade requires you to do the full uboot and kernel image upgrade as described here: https://www.arduino.cc/en/Tutorial/YunUBootReflash
This procedure cannot be done remotely: you must have a USB serial connection to the Yun and be running the YunSerialTerminal sketch. Unless your Yun is a rather new Arduino Yun Shield, the odds are almost certain that if you try to just apply the system upgrade through the web interface, you will make your board unresponsive and inaccessible through any type of network connection - WiFi or Ethernet, using web, SSH, or whatever. If you have an actual Yun (not a Yun Shield) the bootloader on the board is probably incompatible with the 1.6.2 image, and you must upgrade the bootloader as described in the link above.
It sounds like you've already tried to apply the upgrade using the web interface. In that case, Linux is now not able to boot, and you will not be able to make any kind of network connection. I think your only recourse now is to reflash using a USB serial port direct connection, following the directions in the link in my first paragraph above.