There are 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 a pick up from the airport to the hotel, or you can hire a cab. If you want to travel by land then there are several buses leaving from Kathmandu and some have really good service. 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.
Though it is not a tourist hot spot, you can see Lord Buddha stupa on a tiny hillock called Vishwa. Apart from this, Gadhi Mai is a well known Hindu goddess temple which holds a mighty fair every five years. Hindus from all over the world come here to sacrifice animals to show thankfulness to the goddess. Animals sacrificed on this occasion are one of the highest amounts in the world. To the west of Birgunj, Bindawasni mai is another historical and religious place. Jang Bhandur Rana, the first Rana prime minister held a massacre in which hundreds of people died as a result of a royal conspiracy. To the north of the city there is Parsa Wildlife Reserve, one of the biggest national parks, having hundreds of endangered species. Tigers, elephants, musk deer, Gharials, Red hornedbills, etc. are the main attractions.
If you're looking for a local Ncell SIM card, Birgunj has an official Ncell store halfway up the main road on the right from the border as you walk towards the large tower in the middle of the road. They speak perfect English! For more information about telecommunications in Nepal, have a look under the 'Contact' section in Nepal.
There are very few 'restaurants' in Birgunj. Most food is street food, but there are restaurants serving a variety of western, Indian and Nepali food attached to most hotels in the city.
There are many takeaway alcohol stores on side streets, a block either side off the main road. Otherwise hotel restaurants serve alcohol. If you feel thirsty then on almost every 'chowk' or a nearby place there is a hand pump placed for ground water. Most of the people here rely on this water for all purpose from drinking to washing. It will be safe enough to take a sip or two.
Most hotels are located on, or a few meters off, the main road.
An abundance of mosquitoes in April, together with heat of 40C, make sleep almost impossible without a good fan in early April.
To get to Kathmandhu, there are options of traveling by land or by air. Simara airport, located 25 km north of town can be reached by a cab in 25 mins. Many flights operate daily for NPR 2000-2500, taking only 15 minutes to reach Kathmandu. One can also travel by private bus or jeep (sumo). Buses leave 6am-10am, then 7pm-9pm, with many 'deluxe' bus having A/C leaving daily at 7pm. Sumo's/Jeeps's can usually be taken from 6am-3pm, and cost Rs 500 NPR per person in a ten person shared Jeep, or Rs. 5000 NPR for a whole private jeep to yourself or your traveling companions. To get either a bus or jeep, you'll need to go to the bus station, which is located just off the main road. Keep going north (3km from the border) until you reach a large tower in the middle of a round-a-bout called Ghantaghar (literally Clock House). Take a right hand turn and the bus stop is 500m down that road. While it's little more than a dirt patch in the north of town, you'll know you got there as there are always plenty of buses parked. A cycle rickshaw should be about NPR 80 for the distance. Jeeps leave when full, from near the road (not towards the back of the bus stop) so expect to wait up to an hour.
For information to cross the border into India, you may find the information under 'Get out' under Raxaul useful.