Heading the long journey north to Bir Moghrein is mainly for experiencing the landscape - hence you should avoid the over-nights journeys, and try to arrange for a journey from sunrise.
The landscape of this part of Mauritania, Ghallamane, is dominated by numerous and enormous guelbs (crater-like depressions) framed by extremely ancient rocks.
Bir Moghrein itself is little more than a military outpost, and there are virtually no attractions for the traveller in town.
Eat and Sleep
Nothing, go for camping and bring your own provisions.
Shop around in Zourate for private transportation. Expect to use time to get here, and even more time to get a ride either to Algeria (more difficult) or back to Zouerate.
440 km south: Zouerate
250 km northeast: Oued Ben Tili
900 km east: Chegga
600 km northeast: Algeria (Tindouf)