A beverage cooler is a compact refrigerator, often with a glass door, so it follows fridge rules with extra care for the glass. Empty it, defrost and dry it, protect the door, and give the compressor upright rest before you power it on.
Pro tip: If it is a thermoelectric model with no compressor, the rest time matters less, but still protect the glass.
Watch out: The glass door will crack under impact, so bubble wrap it and load the cooler from the side.