I am a Nigerian citizen and presently hold a Russian internship (Стажировка) Visa. I intend to obtain a Russian study visa from the nearest country to the Russian Federation. Can a Nigerian citizen obtain Russian study visa from Abkhazia?
If you're residing in Abkhazia, try to contact the Russian Embassy in Sukhumi so they can provide all the information you need, and the list of documents you'll need as well.
phone: +7 840 2263693, 2260491
fax: +7 840 2265693
address: Sukhumi, Lakoba st, 45
However, getting into Abkhazia is quite tricky (as it's not an independent country, strictly speaking), and you need either russian visa or georgian to get into there, as no other countries has any travel connection with it.
Update: missed the part about your student visa, sorry. So, for obtaining visa to Abkhazia (which you need) you need to fill the electronic form for it and send it to embassy. Usually it takes up for 7 days to get the permission to enter the Abkhazia (take a note that this is a crime in Georgia for you to enter Abkhazia from Russian side). However, the Russian Embassy may have additional information - maybe you can apply for continuation of your visa without going to other country.