The wheel covers press onto the rim of a steel wheel. They have plastic gripper mouldings which are held under tension by a steel ring, the thickness of a coat-hanger.
So the wheel only has to be a regular road wheel of the same diameter as the cover, and the cover will fit.
What size is the "spare" wheel you bought? It looks like 15" and "4-lug" (what you actually need is a 4x98 PCD) but I can't see the width or the offset (ET ) number.
If the width and offset are the same as your existing wheels (the width and offset are stamped into the rim of your old wheel.. though it may be under some dirt or rust) then the wheel is the same size, and no problems.
If the wheel is a "spare" wheel... i.e. is much narrower and has a different offset to a standard wheel, then it could be a space-saver (to fit into the boot) and the face of the wheel may project too far for you to fit the cover. BUT you should not fit a cover to a space saver - this is a spare wheel, only to be used while your regular wheel is being repaired/replaced.
From the advert, the wheel looks too wide to be a space-saver, so it seems they use the word "spare" to mean "additional", not actually "spare" as in "the wheel in the boot which you only use when you have a puncture" etc. so you should be good.
Ralf S.