The distance between the city of Goma in the Democratic republic of Congo and the Boarder of Rwanda is relatively short. It takes about 3 hours from Kigali by road to Gisenyi on the Boarder in the Rwandan side. From Gisenyi to Goma is just a few minutes and you are in the city center.
Mountain Nyiragongo is found in Virunga national park which is also accessed through the same route. From Goma town to the starting point of the hike is about 15km. It will cost you about USD20 and you may choose to use a Taxi or motorcycle. There is an option of public means by buses but usually these are crowded and it had to find a seat so you may end up travelling while standing.
Therefore if you are going to Nyiragongo and most of the Virunga Mountain you will require to use the Kigali airport since it is nearer and then you will travel by road to the Volcanoes. The other major airport of DRC is far and Uganda is another option but that would require if you have a continued safari right from Uganda. You can also fly from Kigali but at times this is unnecessary as there is a lot of adventure you will miss on the road and yet the drive is not long.
The hike to Nyiragongo starts usually at 10am and the easiest thing is to spend a night in Goma but there is an option of staying in Gisenyi and cross the border in the Morning and drive straight to Kibati the starting point. This last option is however tricky especially on working days as the clearance line can be long especially on working days.