If you are keeping/using the head or offal, trotters, flare fat etc., I'd agree with HH a standard fridge would be insufficient space for one pig. If that's all you've got, start with cutting the carcass into the large joints - legs, whole shoulders, belly and loin, that way all of the crucial meat can be kept cool in the fridge almost from the start, then work on the individual bits one at a time out of the fridge, starting with the loin, as that is too long to fit in a fridge whole. We were lucky and had a spare pig length chest freezer which we kept at cool fridge temperatures on butchery days. Could you borrow a spare fridge from anyone?
Also be very well prepared for packaging and labelling which takes far longer than you think, especially when the meat has started seeping blood after very little time in the fridge, and also what you will do with all the waste, as there is a huge volume of that too. If you are short of space generally, the best thing is to get as many joints as possible into the freezer as fast as you can, with temporary packing and labelling if that helps speed things up.