Update: made it safe and sound with no problems at all.

The trick is to arrange an "airport entry permit" -- in advance -- instead of a visa. This then allows you to get a visa on arrival, regardless of nationality, and with only a short wait when you get there.

I had applied for the visa several times with no success, waiting weeks without word, but the airport entry permit took just 3 business days.

I worked with the guy from this agency -->

http://www.lenditravel.com

And I stayed at the German Guesthouse, a nice place that I would also recommend -->

http://www.germanguesthouse.com/vn/hotelbooking.aspx

(I have no relationship with either of these groups, aside from being a paying customer. As always, YMMV.)

Khartoum itself was fun. I did a city tour, went to the national museum and a couple of mosques, visited the markets, etc. I actually wished I was there a day or two extra, which is unusual for me.

I came in on FlyDubai (easy and cheap) and left two days later to begin the AONE4 on the only RJ flight of the day. I was connecting back to DXB and had a long (6 hr+ layover) in AMM. If you ask for it, J class pax get a free voucher for a day room and meal at the Golden Tulip -- but it seems you have to know to ask, because this information wasn't volunteered.

Anyway, it took a while to make it happen, but I'm glad it worked out. If anyone has any questions, I'm happy to help if I can.