The confusion about finite or infinite time dilation at the Schwarzschild radius began since Karl Schwarzschild in 1915 used a space-time coordinate system where coordinate time went to +∞ when a falling object approached the Schwarzschild radius. But t → +∞ for a world line that approaches the horizon is a thing different from t → +∞ far from the black hole. An object that is about to cross the horizon is in no position to synchronize clocks with an outside world.
No, completely opposite situation to the OP’s expectations. If a falling observer will transmit a powerful beam of light out of the hole, then photons will reach increasingly far future of the exterior universe. But the future seen by the falling observer will be very limited; see http://jila.colorado.edu/~ajsh/insidebh/penrose_schw.html for graphic depiction.