Take it outside, stand it on end, and beat the ever lovin' clack out
of it. Then vacuum over both surfaces real well with a vacuum.
If you want to get the internal dust out, you will need to make a
small slit in the edge at the foot and stick your crevice tool in
there and work it back and forth.
This is assuming a spring mattress with only batting on each surface,
rather than a solid cotton layered mattress.