Update: Georgian Railway still (2016) offers free Wi-FI (at least in Tbilisi - Batumi line), but it is slow, because most of the passengers have tuned their devices (laptops and mobile phones) to online video streaming services making this a very bad internet browsing experience.

Once I noticed a dude opening torrent sites and downloading movies (most probably piratical)  - while browsing YouTube meanwhile.

For some serious work - Wi-FI offered by Georgian Railways (it' s Geocell actually) - is not an option - and if you need a stable internet connection while on trip - I highly suggest using 3G/4G - personally I'm using Magti 3G internet which is - cheap and fast most of the Tbilisi - Batumi route - there are places enroute where no mobile networks are available and Wi-Fi neither.