Can the Pont du Gard be described as both an Aquaduct and a Bridge?
It was built originally as an Aquaduct of course, but has served as a bridge (to transport other things than water) in the past. Or is this common among aquaducts?

No it isn't. To start with many aqueducts don't cross rivers on their way.

I wouldn't have many qualms about listing it as both.
However in a slight of hand there is a bridge built directly infront of the lower tier of arches. This is what people walk on now but from a distance you can't see it as it matches the lower part of the aqueduct. So technically the bridge and aqueduct are seperate things.