score:1
No, there is no other public transport to the airport that early in the morning.
The earliest marshrutky leave Kutaisi around 6:00 (source). The only option earlier is the Georgian Bus shuttle or a taxi.
Upvote:3
Georgian Bus company makes transfers from Kutaisi city to Kopitnari (Kutaisi airport) and vice versa. Usually they adjust the departure times to certain flights, so you'll probably be satisfied with their schedule, depending on your flight.
Also, when I was flying from Kutaisi in 2013 I used taxi-bus (marshrutka) Kutaisi-Samtredia which departs from Kutaisi Bus Station (look, location) regularly, however, you should drop off on the road, as taxi-bus doesn't stop directly at the airport.