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Visa policy of Ethiopia on wikipedia
You can get an Ethiopian visa in Khartoum quite easily. The embassy is located on Amarat Mohamed Nejib Street 11, in front of the big cemetery. You'll need a copy of your passport, your passport, 2 pictures, and 40USD, or 60 (if you apply for a three months visa). You cannot pay with anything else than dollars if you are not Sudanese, you can get dollars on the black market from some people or some shops, ask around in the shops or so. The visa lasts one month(or three,which you get just as easily,the only difference is the price) starting when you get the visa in Khartoum, which is not very practical.
It is a mountainous landlocked country. The capital is Addis Ababa.