Instead of the 16" wheel with standard 185/55R16 tyre, for the spare get a 15" steel rim and fit 175/65R15 tyre ( which are much more common and a lot cheaper ) - my wife has 15" alloy wheels on her MK2 ( the original 16" alloys now in garage ) for a few reasons, it was gonna cost more to get the original Si diamond cut rims refurbished ( they got 'white worm' ) than buying new Rial 15" alloys + Avon tyres, the ride is better with deeper sidewalls and less chance of pothole damaging the rim, less noise, and much cheaper tyres with a much better selection. She has one of the original 16" wheels in the wheel well as a spare.
The rolling radius of 185/55R16 and 175/65R15 is within about 1%. ( in fact both of those tyre sizes listed on the homolgation sticker on drivers door 'B' pillar, below the door lock plate )
This is a very useful site, it cross references different car wheels with other cars, so you can go to scrapyard ( if you want to ) or Ebay and get a cheaper rim just for spare.
I would go for full size spare any day over the wheelbarrow space saver - having driven on spacesaver a couple of times it is not for the fainthearted. Those spacesavers are expensive anyway, and the 2008-2015 Jazz will definitely take a full size spare, as others say just through the gunk away ( it goes out of date anyway ) - shove the expanded foam insert into garage and shove a full size wheel in the boot, you can buy the holding threaded bit, but I made one from a piece of M8 allthread, a bit of 2mm thick steel plate ( buy a bracket from B&Q or Wickes and cut it up ) and an M8 wingnut - easy peasy.
or this
https://www.coxmotorparts.co.uk/honda-shop/genuine-honda-bolt-anchor/Get a jack and wheel wrench if you haven't got one
https://www.coxmotorparts.co.uk/honda-shop/draper-jack-wrench-set/https://www.wheelfitment.eu/car/Honda/Jazz%20(2008%20-%202015).html