If the earth was flat, would the transition between day and night be sudden?
Well, assuming a lot of things aren't true, then no. For example, if the Earth was shaped like a pizza, and the sun revolved around it, then it would have the same transition.
Depends on the size of the sun.
If it's capable of illuminating the entire surface of the earth at once, then the transition would be sudden. Take a torch, and a coin, and sit in a dark room. Shine the torch obliquely onto one face of the coin. You can then choose whether the coin spins so alternating faces are exposed to the torchlight, or the torch orbits the coin, achieving the same effect. It quickly becomes obvious that the transition is quite sudden, no matter how slowly the coin moves. Do the same thing with a ball or a marble, and the transition is gradual.
However, if the sun is only capable of lighting a small portion of the surface, like a spotlight, then the transition could be gradual, as the beam sweeps about the surface. The Flat Earth Society even has a snazzy graphic about it here. Of course, that raises all kinds of other issues such as whether one could construct a path that exactly reproduces the observed day/night transition times, but they seem to be only tangential to your question, and if we're arguing about what was observable from the Middle East in 700CE, that makes it even less relevant.