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Have a look at this link. You apply for visa online, and then physically get it by reporting to a consular service within 3 days of entering the country (you get an entry permit sent to your email which you use to actually enter). I would not advise trying to obtain visa/entry at the border, unless you are exempt from visa by Abkhazia.
Also take note of the following important info:
Citizens of the following countries must have double entry or multiple Russian visas to come to Abkhazia (it is a mandatory requirement, since the Russian Embassy in Abkhazia will not issue transit visas to the citizens of these countries):
Algeria
Angola
Afghanistan
Bangladesh
Vietnam
India
Iraq
Iran
PRC
DPRK (North Korea)
Nepal
Nigeria
Pakistan
Rwanda
Syria
Somalia
Chad
Sri Lanka
Ethiopia
and:
NOTE! When visiting Abkhazia through the territory of the Russian Federation you must receive at least a double entry Russian visa (the first time you enter Russia on the way to Abkhazia and the second time you visit Russia on your way from Abkhazia back to your home country).
Foreign citizens are required to have a double entry or multiple visa of the Russian Federation.