score:4
Yes, taxis are available 24/7. A simple Google search revealed:
http://www.tbilisiairport.com/en-EN/Transportation/Pages/Taxi.aspx
Also:
http://www.info-tbilisi.com/usefulinfos/taxi/
You can book at least one service online, but it appears to be just the equivalent of a paper form.
Perhaps your best bet is this: