You can get a Saudi transit visa in Khartoum, though it might sometines need a second try. Mine was rejected first for no apparent reason and only approved after talking to the consul.
The consulate staff will send you to an agency nearby to do the paperwork. They do the application through a special software. Khaled Medawi of K.M.Z. was very helpful, though quite expensive (I can't really compare the rates but his was 400 Sudanese Pounds which equals more than 300 Saudi Riyals while the stamp on the visa indicates a visa fee of just 50 Saudi Riyals). The K.M.Z. office is in the same side street of the consulate's entrance, some 50 meters on the right hand side (see pic of entrance below).
The procedure usually takes two to three days, so if possible arrive early in the week avoid the weekend!
You definitely need to have a LHD car. The consul and the agency ask this very clearly!
You also need an onward visa, i.e. Jordan or UAE (if you want to continue through Iran). Your passport needs to have at least one free DOUBLE PAGE (one page for the visa, one for the stamp of fees).
Couples travelling together need a marriage certificate.
All the rest is the usual (2 passport pictures plus copies of the car papers, the carnet, your passport, etc.). The agency will tell you exactly.
The visa very clearly states the country of destination, though no precise border crossing. So make sure it is the right one!
Greetings from Khartoum