The holiest site in Sudan is located across the Nile from Khartoum. Omdurman makes a nice day trip to see the site of the Mahdi's Tomb (nonbelievers aren't permitted to enter); Khalifa's House (relics of the Mahdi and British Sudan); and a huge, fascinating market (excellent silver jewelry).
Don't miss the Hamed al Niel Mosque (Sufi "whirling" dervishes dance there an hour before sunset on Friday, except during Ramadan). Omdurman presents quite a contrast to modern Khartoum. At the base of the Nile bridge, boat builders can be seen plying their trade. We consider Omdurman a must-see.
6 mi/10 km northwest of Khartoum.