There is some airlines flying to a place called Simara from where it takes around 45 minutes to reach Birgunj. If you have hotels booked then there is possibility of pick up from the airport to the hotel or you can hire some cabs. If you want to travel by land then there are several buses leaving from Kathmandu and some are really good services. There are also night buses leaving from kathmandu to birgunj and these are the best buses running in Nepal.
KATHMANDU CHITWAN POKHARA DAMAN