A shinto rasp is a great neck carving tool, although it borders on too big and too clumsy for uke sized necks, and it can't do fine curves since it's flat and really broad. It can remove material quickly but it can also leave a fairly fine surface, you just need to get used to the technique - how to manage your stroke vs the grain of the wood, knowing how much to slip vs push across the grain.
A step up from a shinto rasp would be Iwasaki carving files. They cut quick like a rasp but leave a very fine surface. They're available in a variety of shapes. The half round shapes are great for carving around headstock transitions and heel blocks. My ultimate favorite hand tool for any sort of carving or adjusting of joinery or other similar tasks.
Pull saws are nice too. I will second what everyone above has said about them. But the best news is that even the cheapest pull saws are pretty good. My favorite pull saw is an $8 saw from Harbor Freight. I use that saw more than any other hand saw in my shop. probably by a factor of ten to one.
If I were in your shoes and considering the tools you've mentioned, I would skip the shinto and spend more on Iwasaki files. And get a cheaper pull saw instead of the $30 one you linked, to balance the budget. If you can only afford one Iwasaki, get the 6 inch half round fine cut. If you can afford two, get that one and the matching 6 inch flat fine cut. The half round is great on heel blocks and headstock transitions. The flat is great to adjust joinery. Either can be used to carve the length of a uke neck but the flat is a little faster and more steady for long straight carving like that.
Here's the iwasaki files:
https://www.highlandwoodworking.com/iwasaki-carving-files.aspx
The naming conventions are confusing, if you cross-shop, make sure you're comparing like for like. Sometimes the names include lengths in inches, sometimes in cm, and sometimes those lengths are overall lengths but other times it's just the cutting length. There's only a handful of distinct sizes and shapes but they get called all kinds of different things at different vendors.
Here's the harbor freight saw I'm in love with:
https://www.harborfreight.com/12-in-flush-cut-saw-62118.html