As far as I know, no.
To enter Georgia from Russia, you'd have to go through Abchazia, which has not been under Georgian control since 1993. The security situation there is precarious, there are several unmarked mine fields. Abchasia is supposedly closed to international travellers. Citizens of CIS may enter, I heard, but not others.
I think your only option is the ferry from Sochi, RU to Trabzon, TR.
At least, that's the current state of my research. If you find out otherwise, let me know and post it here.