How do I remove the automatic update

I don't think you can remove an automatic update. It should not be removed as these updates are critically important. And, as noted, if you do not address this issue, your computer will stop functioning at some point. Clearly, you need more storage. That is easily accomplished by installing an external disk. These days they are reasonably priced, compact, very easy to install with a provided USB cable and most do not even require external power. Sometimes you may have to reformat it, though. Once it's installed, just copy files over on to it and then those files can be deleted from the computer's hard drive. Problem solved.

Totally agree with colleague @Ronasara - do not turn off Auto Updates.

Re-reading your original posting - the cause of running out of space has nothing to do with the Big Sur updates. It does totally depend on the Users' Management of available Space. Only the User downloads, installs and keeps large Applications, Videos, Movies, Music etc. As @Ronasara has suggested - get an inexpensive external drive usb powered and starting moving unneeded things off the internal and onto the external drive asap.

Strongly suggest making space before the computer will stop functioning at all. If this happens - you might have to use Target Disk Mode to create space just to boot up. What is “Other” storage on a Mac, and how can I clean it out? How to free up storage space on your Mac

How much free space do you currently have on your boot drive? As P. Phillips pointed out if you get too low in free space the MBA will lock up and you won't be able to boot up.

If you need to off load files but carry them with you this is an excellent companion to laptops: OWC Envoy Pro mini

We strongly recommend you IMMEDIATELY free up at least 50 GB of space, more if possible, on your MBA to eliminate the danger of locking up. Also it will help improve overall performance from the system and apps.