The AE71 Corolla were available in Auto. 4AC engine, so no problems with bolt up. From memory, they were not computer controlled auto's. However you could adapt a TA22 Celica Auto box, as they were definately not ECU controlled. A 1G-GTE early version Auto box had an independent/stand alone ECU for the box.
Totally agree withe the comments regarding longevity of the box,without the mentioned mods. You need to have quick gear changes, otherwise the clutch plates will slip big time due to the torque of the 4AGZE.