score:9
I tried looking at the biographies of known native Africans. An obvious starting point is Nelson Mandela, he went to the University of Fort Hare. Close but no cigar: that university was founded in 1916. However, his biography also mentions University of South Africa (founded as University of the Cape of Good Hope in 1876) and University of the Witwatersrand (founded as South African School of Mines in 1896). Mandela studied there half a century after the period you are asking about here but if one is to believe Wikipedia on this, there were no acceptance restrictions for black students before late 40's of 20th century. So at least theoretically native Africans were allowed to study there. To prove that it also happened practically one needs to actually find somebody who studied there during the period in question (e.g. Sol Plaatje had to resort to private lessons, no good).
Looking at personalities from Ghana, things first looked worse there. I could find lots of people who studied in Europe: Germany, France, England. For example J. Benibengor Blay who studied at the Regent Street Polytechnic in the 30's of the 20th century. Or Jacobus Capitein who studied at the University of Leiden in the 18th century. Then I found J.E. Casely-Hayford, looks like he attended Fourah Bay College in Freetown (Sierra Leone) somewhere around 1890. Seeing that this university existed since 1827 it must have been an obvious destination for native Africans looking for an education option. Africanus Horton and Samuel Ajayi Crowther attended the same university earlier. One of the sources for the Wikipedia article even says:
From 1827 to 1950 and from 1969 to the present, the majority of the faculty was African.
I looked some more but couldn't find any other university in West Africa, maybe Fourah Bay College was the only one at that time.