The refrigerator is the way to go. You have the temperature range right, but it needs that continuously for 3 months or longer, until you are ready to have it begin growing in spring. Warmer periods could prevent it from getting the winter chill it needs to break dormancy properly, or start it growing weakly before you want it to. If the foliage has (nearly) died down it is ready for dormancy.
The natural soil you have it in will be too heavy and stay too wet in a pot. Potted Cyps do best in a gritty mix that stays aerated, think sand and perlite and a little peat moss, or something like bonsai potting mix, or do an internet search. You should be able to find some specialist vendors that sell Cyps and supplies. A little of the original soil mixed in might help establish the symbiotic fungus that Cyps always have in nature, but the fungus is not necessary for pot culture.
Some of the hybrid Cyps are easier to grow, but Cyp pubescens is certainly among the easiest of the species.