I have had that happen and so long as no water got in the gas, if it did you would have to drain it and change the fuel filter, or there is no water in the carb and the filter to the carb is not saturated with water, once everything dries out it should be OK providing nothing rusted while it was drying out.
No, it will be fine, check the gas as @ Hypocrisy Central mentions and then just wipe it down, oil anything that might need it and drop the paranoia! haha
If the lawnmower engine was really hot from mowing a large lawn and left out in a sudden cold downpour shower, then it’s possible that the cold water could crack the hot engine.
Otherwise the lawnmower should be no worse off from one rain storm.
I often hose down my lawnmower after mowing the lawn but I wait until the engine has cooled down.