Whose house is next to Nabstablook's house?
Solution 1:
The house next to Napstablook's belongs to Mettaton.
You can buy the Mystery Key from Bratty and Catty for 600G.
Also:
in the True Pacifist ending, you can wander around the game world after beating the last boss. Mettaton will be found hanging around outside his house in his Mettaton EX body.