You don't convert the car to OBD2, you convert the motor to OBD1 and run a jumper harness from the ECU harness to the new OBD1 ECU.
If you stay auto trans you'll need to get a 93-95 JDM Integra GSR auto ECU. That is the only OBD1 Bseries Vtec auto ECU I know of. Though you could convert another ECU to work if you have the right parts and knowledge I suppose. You'll still probably want to have it socketed and have it tuned just to make sure everything meshes.