It is not difficult to replace individual asbestos sheets, you will need to loosen the overlapping adjacent sheet in order to fit the new one under it and make sure that you get the head of the sheet over the relevant purlin and under the sheet above.
If you take a piece of the broken sheet along with 1 of the fixings to your supplier he will sell you a non-asbestos fibre cement sheet with the same profile, make sure that you measure the full width & length of the required sheet first.
You will find it easier with a second pair of hands but it can be done solo.
As for disposing of the old asbestos, many councils will accept small amounts from the general public if they have a special skip on site so it is worth ringing them.
Doug D, March 2011