Is it possible when the Universe begins to fall in on itself that the arrow of time will reverse itself?

It's not just General Relativity that prohibits time travel, even the solutions to Einstein's Special Relativity show that time cannot go backwards. A lot of science fiction writing seems to think that if you simply go faster than the speed of light, then time will start to roll backwards. Well, the solutions to SR put an end to that plot element right away. When you set v > c, time doesn't become a negative number, it actually becomes an imaginary number (so does length)! So the solution to going faster than light is not time travel, but something even stranger: you leave time and space entirely, and enter some other domain!

Another way to look at it is let's say you're in another part of the universe where the direction of time flow is exactly 180 degrees reversed from the direction of time flow in the part of the universe we are currently in. When you are in that alternate part of the universe, the flow of time will look like it's flowing forward for you there. It's pretty similar to if you're in Australia and you look up, and you're in Canada and you look up. The two directions of up may be 180 degrees apart, but for the locals there it's still up.

No. There is a solution to Einstein's general relativity at the scale of the universe. In it, time is always proceeding in a positive direction. The spatial component of the solution has a scaling factor that is either positive or negative which corresponds to either an expansion or contraction of the universe, but in both cases the time component is always positive. There are (of course) some simplifying assumptions concerning the distribution of matter and energy (homogenous and isotropic), but it should be quite accurate at large scales.

No. When the Universe begins to fall in on itself it will fall into a "black hole" and never reverse itself and never will return. E-mc2

Steve Hawlking used to think that but now he changed his mind...

If it did, many physicists would be very surprised.