The newest bridge linking Khartoum with Omdurman is the Chinese Bridge, an imposing structure which is amazingly steep and high. On the Omdurman side, there is a long walkway by the Nile which has now turned into a pleasant place to spend an evening...some enterprising locals have even set up a stall selling drinks and nuts, and a few chairs and tables have appeared. The only difficulty is getting there and getting back, as not all buses between the two cities use this bridge, and it is difficult to flag down a bus on this stretch of road. It is also quite a distance from the centre of things in Omdurman, and is not a pleasant walk...but if you have transport, then it would be a nice place to spend an evening by the river.
Omdurman isn't exactly a hip and happening place at night, although there are a few things you can do at night. If you are male, you could go to a chicha "den" for a pipe, or watch a film at a cinema, but an alternative for the whole family is to walk the Shambat Bridge which links Omdurman with Bahri (North Khartoum). Most evenings around sunset, people gather to take in the views, mix (a man and a woman socialising together is called "mixing" in Sudan), and take exercise. Up until about 9pm, you can see whole families walking the length of the bridge, couples talking secretively, and fitness freaks jogging and doing press-ups on the pavement. Don't go too late, as this bridge also tends to attract the mischievous, and your presence might arouse the suspicion of the police, but before 10pm it is definitely the place to be seen.