Partner's pretty much correct. While it's not drive cycles, per se, there are individual criteria that need to be met for each test. Unfortunately, Hyundai doesn't publish which parts of the readiness cycle are necessary for each test.
Here's my take on your specific case:
Fuel system: This isn't generally a specific test. I suspect they're referring to the evaporative emissions leakage test. On older models, this test would only run when idling for 30 seconds or more. Your dealer can command this test to run.
Oxygen sensor heater: This test usually runs fairly quickly after startup. Since you don't mention oxygen sensor performance, I suspect this is the test that actually hasn't run. In fact, the performance test cannot run until the heater test has run and passed. Often the car must be driven at constant highway speed (50 - 60 MPH) for several miles for this test to run.
Catalyst: This test cannot run until the oxygen sensor test completes, since it depends on oxygen sensor data to determine catalyst condition. In fact, this is probably the most difficult test to cause to run on your car. This test will require several miles of steady speed highway driving after the oxygen sensor test has completed.
You can purchase (not sure of price or whether it's worth it) a code-reader which will also report the status of the readiness tests. That way you'll know before you take the car in (and pay the fee) whether it'll pass.
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