Welcome to Nikonians! Yes you will have the same problem on the D90 or any other Nikon DSLR in the Vari-Program (pictogram) modes, Auto mode, or P,A,S modes. These modes rely on i-TTL flash control.
You can use the SB-24 on the D5000, D90 or any other Nikon DSLR with the camera set in Manual exposure mode and the SB-24 in Auto (non-TTL Auto) flash mode or Manual flash mode. The calculator built into the Speedlight will show you the correct Aperture to set based on the flash to subject distance. Good Luck and Enjoy your Nikons!