Jonathan, as far as I am aware, EEC citizens enjoy visa-free entry into Georgia.
See:
http://www.mfa.gov.ge/index.php?sec_id=467&lang_id=ENGYou can fly Georgian National Airways for about €145 return which is, of course, a little pricey for a one day trip. I cannot see how you can do it in a day since the flights depart KBP at 17:25 hours. The return flight is always two days later at 15:15, necessitating a 2 night stay.
I have not tried any other airlines for both tariffs and timetables.
If you do go - wonderful people, great sights and superb local cuisine! Enjoy!