Is it possible to mount an 03 dually bed on a 99 truck?

Anything is possible. You are only limited by your creativity.

I have seen 1st gen with a 4th gen bed so I'm sure it could be done.

I remember a friend says it's bolt on.  Take it for what it's worth, which, in this case isn't much considering we have "guy on internet" "memory" and "a friend" all in the same equation.