You could also try a primer coat of paint that is designed to bond with plastic, like Krylon fusion or the Rustoleum equivalent in addition to washing and possibly sanding. I don't think they sell Krylon Fusion primer specifically, but you can just pick any color and use that. Flat would be ideal if they make it. I haven't tested this, but it should stick well to the plastic nose cone and provide a good surface for other paint/primer to stick to. It might not work as well with airbrush paints, though, since they are mostly waterbased and don't have solvents to make a good chemical bond with the Krylon Fusion paint.
Another option is Duplicolor Adhesion Promoter, which is a clear primer composed of mostly solvents that you spray on the nose cone before painting with regular paint, but again I think the second coat is meant to be lacquer/enamel-based spray paint, not water-based airbrush paint.