Is it reasonably possible to currently travel from Juba to Khartoum (or vice versa) over land? How? What would be the route? What would be the transport?
Considering recent border conflicts I would suggest you check travel advice from your government on this issue. My guess is that it would be best to leave it for a year or two. South Sudan is a new nation in the process of building up its government offices. For one border stations might not be properly set up. I guess getting a VISA on the border might be difficult. Taking vehicles across borders could in itself be a challenge.
Roads are of course subject to general quality and seasonal changes. A shallow search on the net indicates that the rainy season may make it near impossible to travel the stretch.
In a discussion on the Lonely Planet webpages back in 2008 someone asked the same question as you:
Basically - it sounds like a bad idea. But one of the contributors in the above hiked from the south to the north. Quite recently it seems. The person also mentions that parts of the trip might be done by boat from Malakal to Juba.