Khartoum is not a city of grand monuments, but it has a handful of genuinely interesting sights. The Sudan National Museum, with its Kushite collection and rescued Egyptian temples, is the obvious starting point. The confluence of the two Niles is the other essential — best seen from Mogran Family Park or from the Tuti Bridge.
The colonial downtown still has its Republican Palace, the small but beautiful Greek Orthodox Church, and St Matthew's Cathedral, all walkable in a single morning. The chaos of Souq Al Arabi is good for a sense of how the city actually runs. For something quieter, the Khartoum War Cemetery and the Sudan National Botanical Garden are reliable refuges from the heat.
The other essential half-day is across the river in Omdurman: the silver-domed Mahdi's Tomb, the vast Omdurman Souk, and the shell-shaped Al-Nilin Mosque make a logical loop.