You need to think about where the water comes from and where it will go. If the lowest point is behind the shed then that is where you need some sort of drain. The water will be draining down to that point from the rest of the garden so the relative contribution from the shed roof will be small.Collecting rainwater is always a good idea though so guttering and a water butt is worth doing but it will have limited impact on your winter water problem.
If your shed wasn't there you would still have to deal with the same volume of water, no more and no less. All your shed roof does is delay the arrival of rainwater to the ground by a few seconds at most. The problem is that however the water gets to that corner of the garden it isn't draining away fast enough to prevent surface flooding. Are your neighbour's gardens also flooded? Do their gardens drain into your's? Although you are on a slope you might still have a dip in levels which allows water to accumulate.