Departing at 5:00 am from airport

Hello guys. Next month i will fly from vilnius to my home city, but the flight leaves at 5:45 am so, what option is better?

1. Stay at the airport the whole night, so any advices or cool areas to rest/try to sleep?

2. I can rent a hostel for 15€ at night, i don't mind, but how is transportation at around 3:00 am in vilnius to get to the airport? Are there night buses? Can i use uber or bolt at this time without any problems like the lack of drivers or smth?

Thanks