New Vegas' lights come from power from Hoover Dam. Initially, the power you see is likely "Bleed" off the 5% of the Dam's output that goes to the Strip.
In the Capital Wasteland, given how vital D.C. and the surrounding area was, I simply assumed the fission and fusion reactors were still active. The car's explode in a micro-mushroom cloud when they take enough damage, so they're obviously still functional. So too would any underground reactors.
As for Robots, they have their own internal power supplies. Obviously, these power supplies are extraordinarily long lasting. Though if we're going to get into the realism aspect, two centuries of neglect would have rendered most robots inert due to simple lack of maintenance.