Is a minimum of 1000µF capacitance needed between an lm386's output and a speaker and if so why?
The capacitor referred in the video forms a high pass filter with the speaker (C-R filter).

Its frequency response is given by the following formula:

$$f_c = \frac1{2\pi \tau} = \frac1{2\pi RC}$$

Using this calculator you can see that a 4 ohm speaker and 1000uF give a "cut-off" frequency at about 40Hz. 100uF would give a "cut-off" frequency at about 400Hz. 

Whether it is audible or not it depends on the music you are listening to and your speaker system, etc.

Notice that I mention "cut-off" in quotes because it is not a hard barrier. It's usually given by a graph like this: