Considering that any punctures I have had have allowed me to pump them up and get to a tyre repairer I've never had the need to use a spare tyre nor the repair gel in over twenty years of driving Porsches but as Ralph says the alternative repair of plugging the tyre is more viable than any other means you could carry out at the roadside.
However, there is an article in the Technical Article Section of our Cayman Forum which covers installation of a spare wheel which will answer your questions.
As for Tyre diameter/circumference all Cayman wheels 17", 18" and 19" have relatively close sizes due to their having different aspect ratios, you will need to check the difference of a spacesaver and ensure that it will clear the caliper, not such an issue on the 2.7 and a full size 19" wheel with tyre will fit in place on the rear parcel shelf. See the article, Porsche have previously used two types of spacesaver tyre, one being collapsible the other being a narrow but fully inflated tyre.
Personally, on continental trips I carry a couple of cans of an alternative gel which is washable and so easily cleaned out of the tyre it connects to the tyre valve and is pressurised so a quick temporary fix is possible.