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.
Travel by Tata Sumo jeep from Birgunj to Kathmandu takes 5-6 hours and is a bumpy ride at best--perhaps not for the faint of heart. The ride should cost INR250-350 NPR 500.
For more details on crossing the border from India to Birgunj, see the information under 'Get Out' on its Indian sister's town page Raxaul.
