They had parking on the back of the building. I used to drive there very early in the morning, don't know about now tho (moved to Samut Prakan a few years back)
If your phone is locked, you can buy a cheap unlocked phone with a Thai SIM and use it as a Wi-Fi hot-spot. Otherwise, go to AIS counter at airport (at arrivals) and they will set up the SIM and data connection in less than 10 minutes. Other companies like True and DTac are also good