Update: Georgian Railway still (2016) offers free Wi-FI (at least in the 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 the 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 en route where no mobile networks are available and Wi-Fi neither.