Can’t remember as I am used to Campag stuff
Removing a 10 speed Shimano cassette to fit a SRAM.
Do I need the little spacer/shim thingy?

If it’s a quite thin spacer then no, it came with the Shimano cassette. Wheels with 11 speed compatible hubs usually have a thicker spacer (about 1mm) that you would need to reuse.

I’m guessing you have just the thin spacer that came with your cassette.

I would say yes. I have nova tec hubs on the road bike with sram 10sp cassette. Without the spacer there was around 1mm of play which led to much noise.

sram and shimano cassettes are interchangeable

all the 10speed/11speed hubs are narrower than 9speed so you will need the 1mm spacer ring if it was already there doesn’t matter if sram or shimano

The spacer on 10 speed Shimano was a road thing. No need for spacer on Shimano MTB cassettes, nor SRAM road or MTB. If you’ve got one it could be hub specific rather than cassette specific, a la Mavic. What hub, and road or MTB?

Hope hubs, Ultegra being swapped out for 1070 SRAM

No spacer needed on the SRAM cassette.