What I want to achive is when given a directed graph, find out if there is one node that can reach every other node and if there is one node that can be reached by every other. Time complexity should be O(n+m).

I'm into finding all strongly connected parts in the graph and then check the connectivity among those. Can that be an appropriate solution or do I have to rethink it all? Maybe you guys have a better solution.

Sounds like a spanning tree problem. I think you invert the direction and go from there, or something like that.