Because melamine was specially crafted to resist kitchen spills like water, oil, and food, it does not take well to paint. That said, a light sanding can remove the melamine surface (be careful you don't sand too much! -- you may also want to wear a mask and/or use a fan to avoid breathing in the dust), and then the surface underneath the melamine will accept the paint.
As always before painting, make sure the area is clean of all debris (dust, dirt, old paint, any oils or grease, etc., then wash the surface with soapy water and let it dry completely.
After painting with Soto's PREMIUM ALL-IN-ONE HOUSE PAINT, it will probably be dry to the touch within about half an hour, but it is best to let it dry for 2-4 hours before using it or adding a second coat. Let the paint dry for 2-4 weeks for maximum hardness and durability before subjecting it to heavy use (heavy scrubbing, putting heavy objects on it, etc.).