Ok, lets get cracking.
Start off by getting a 10mm socket wrench, then pop that Hood
(hopfully your engine bay is cleaner than mine).
Then remove the negetive battery cable.
Then locate the ECU box(it's in front of the battery) and remove the two 10mm nuts.
After that is done, push the tabs on the sides outwards and pull up on that side, the cover will pop off. Leaving you with this.
Remove all of the plugs, and pull up on the blue piece on the left to reveal another 10mm nut. Remove this nut.
The stock ecu should now pull right out. Hook up the A'pexi unit, put the ecu cover back on, reconnect the negettive battery cable, and your good to go. The English manual says to let the ecu calibrate, by which it means you should let your celica idle for 10 minutes without the defrost or AC on. Then idle another 10 minutes with just the defrost on, and another 10 minutes with just the AC on.
And with that you should be good to go, any questions/comments?