Your car is 11-13 years old. Parts are going to need replacing from age. The starting problem you are having sounds a lot like the problems I've had on 2010 Fortes: Difficult to start and seems like fuel starvation problems. If the starting problem is happening once the car is warmed up (recently driven) then the O2 sensor could be the problem. If the starting problem happens after the car has sat overnight then the O2 sensor is not the cause - but should be replaced if you're getting a code for it. From experience, I suspect your MAP sensor needs replacing. The MAP sensor is critical for fuel control - especially when cold and the O2 sensor isn't hot enough to send a signal (ECM running closed loop). The MAP sensor is probably faulty (It's OLD!) but starts to work once it gets warmed up - and doesn't reach the parameters required to set a code. If it were my car I would replace O2 sensor, and if that didn't fix the starting problem I'd replace the MAP sensor with a NEW FACTORY MAP sensor. Keep us updated.
