No, docker will delete the container when it stops, and there's no update option to change that behavior. You can see this in the help output:
$ docker container update --help
Usage: docker container update [OPTIONS] CONTAINER [CONTAINER...]
Update configuration of one or more containers
Options:
--blkio-weight uint16 Block IO (relative weight), between 10 and 1000, or 0 to disable (default 0)
--cpu-period int Limit CPU CFS (Completely Fair Scheduler) period
--cpu-quota int Limit CPU CFS (Completely Fair Scheduler) quota
--cpu-rt-period int Limit the CPU real-time period in microseconds
--cpu-rt-runtime int Limit the CPU real-time runtime in microseconds
-c, --cpu-shares int CPU shares (relative weight)
--cpus decimal Number of CPUs
--cpuset-cpus string CPUs in which to allow execution (0-3, 0,1)
--cpuset-mems string MEMs in which to allow execution (0-3, 0,1)
--kernel-memory bytes Kernel memory limit
-m, --memory bytes Memory limit
--memory-reservation bytes Memory soft limit
--memory-swap bytes Swap limit equal to memory plus swap: '-1' to enable unlimited swap
--pids-limit int Tune container pids limit (set -1 for unlimited)
--restart string Restart policy to apply when a container exits
Trying to change the restart policy will result in an error:
$ docker run --rm --name test-rm -d busybox tail -f /dev/null
14a7d9bfebc3356652e3c3f060e92d9cebb9f3eb4490873540c9eb1c272f6612
$ docker update --restart=unless-stopped test-rm
Error response from daemon: Cannot update container 14a7d9bfebc3356652e3c3f060e92d9cebb9f3eb4490873540c9eb1c272f6612: Restart policy cannot be updated because AutoRemove is enabled for the container
You can try to copy your data out of the container with:
docker container cp $container_name:/path/in/container /path/on/host
If you don't know which files to save, list which files have changed in the container with:
docker container diff $container_name