Is there a shuttle to the airport at 4am?

Private shuttles are usually available 24 hours/day with a reservation. No regularly-scheduled shuttles that I know of at 4 am.

Yes there is shuttle, it might be stopped for few hrs overnight though

There should be. My flight was 7am got to the shuttle at 5am. I'm pretty sure even if the building isn't open, the shuttles will still be running.