Sotchi - Georgia

There are a number of issues. Firstly, you'll need a multiple entry Russian visa. Second, there are no direct buses, you'd need to transport hop. Secondly, the border crossing with Abkhazia is open, but you cannot continue into Georgia (they consider it an illegal crossing) That means you'd need to cross near Kazbegi (Georgia) It's too far from Sochi to be considered a day trip.

As above, the only part of Georgia that is "near" Sochi is Abkhazia. Abkhazia is de jure+ part of Georgia (according to most of the world) but +de facto is an independent entity. So, not only would you need a multiple-entry visa to get back into Russia, but you'd also need to arrange for an Abkhazian visa with the Abkhazian Ministry of Foreign Affairs. This is a lot of hassle for just one single day. And, as #1 correctly states, you wouldn't be allowed to continue into Georgia proper because, as far as the Georgians are concerned, you'd have made an illegal entry onto their territory.

Save Georgia and/or Abkhazia for another trip when you have more time. Mid-winter is not a good time to visit anyway.

Sochi to Georgia won't be easy, not for one day. You'd better find another destination. Sochi is easy to visit and so is Georgia, but not the two combined. There's Abkhasia in between, politically disputed territory.