Transportation options to Kaunas :
Kaunas airport has Ryanair operating with plenty of cheap European flights (see the list of airports connected to Kaunas)
Vilnius airport covering for non-EU flights is in 1 h 20 min by car/bus.
From Vilnius to Kaunas :
Train schedule is available through the link here “Tickets between 10-14 Euros”, the train can go straight from airport (a bit longer to wait) or you can go with taxi from Vilnius airport to Vilnius train station (Avg. 5 EUR and up to 10 mins) and from Vilnius train station to Kaunas train station
Taxi,price from Kaunas airport to city centeraround 15 EUR and it will take about 25 mins only, Taxi Apps are available as Bolt
For Public transportation, you can use Trafi app:
Price 1 EUR, you can by ticket in the bus. It's 45 mins trip to Kaunas rail station. From rail station to Hotel is about 3-4 EUR with taxi (Bolt or Uber)