There are multiple routes to get there. The ground distance from Kathmandu to Mahakulung is about 295 kilometers. You can travel by land and air to get to Mahkulung. You could choose the best option you like.
Mahakulung is easily accessible year-round by jeep from Kathmandu. The distance of paved roads from Kathmandu to Salleri or Nele Bazar is roughly 265 kilometers. If you start the journey by jeep from Kathmandu around 2:30AM in the early morning, you could reach Mahakulung in one day. By doing this, the jeep fare will be less.
There are two domestic airports in the region, named Lukla and Phaplu. You could take a direct flight from Kathmandu to Lukla Airport, the gate way of Mt. Everest. From Lukla, need three days of walking northeast of Lukla to get Mahakulung. Likewise you could take flight from Kathmandu to Phaplu Airport. Once there, you could take a 4-hour jeep ride to Mahakulung. A helicopter is an additional option if you have time limitations.