Theres seems to exist taxis (pick up trucks or old 80's mercedes 200d). I once met a couple of tourist comming from Atar and they told each payed 1500UM (more or less 4 euros) to go from Atar to NKT. So there's probably the same from NKT to Atar. So to Akjoujt the transportation should be the same. remeber that if you're actually going to atar you'll have to stop in Akjoujt to rest for a while. The time you'll rest there will be enough to see what this city has to offer: not much. just enjoy a hot tea.
on the picture you can see how i got to the city. This was a very painful trip due to the extrem heat at the time. Because of the overweight of the car we had to drive 80km per hour, which even made the trip longer.
Akjoujt is 256km from Nouakchott going to Atar. 180km from Atar. Road is very ok. Asfalt all the way.
I think this guy would give you a ride. Is normal to ttrucks to stop and give rides. dont be surprised if someone askes for money to take you. something that might work is before you get into the truck make sure you dont have money at all. say "ANA WALO FLUSS" i have no money... in the end say SHUKRAN thank you.