If you have the engine with OBD2 and the ECU from that same engine with the proper harness I would imagine you could just drop it in and all the systems would run OBD2, but if you have your old OBD1 ECU and the rest is OBD2 I would say all you need is the correct ECU for OBD2.
Also you may need more then one O2 sensor for the correct air/fuel ratio, I believe OBD2 has 2-3 O2 sensors and if you have an OBD2 engine+ecu w/o the proper emissions set up if could cause your check engine light to come on and make your car either run too rich or too lean.

I can't really think of anything else, hope this helps! Good luck!
-OG