If you are seeing the error "W: TMPDIR is mounted noexec, will not cache run scripts" while running the apt-get command on a Debian-based Linux system, such as Ubuntu, it means that the temporary directory (TMPDIR) is mounted with the noexec option, which prevents the execution of scripts on the filesystem.
The apt-get command uses the TMPDIR to store temporary files, including scripts that are used to configure package scripts after installation. If the TMPDIR is mounted with the noexec option, apt-get will not be able to execute these scripts, and you will see the "W: TMPDIR is mounted noexec, will not cache run scripts" warning.
To fix this error, you will need to change the mount options for the TMPDIR filesystem so that it does not include the noexec option. This can typically be done by modifying the /etc/fstab file, which contains the mount points for the filesystems on the system.
For example, if the TMPDIR is mounted at /tmp, you can change the mount options by adding the exec option to the mount options for the /tmp filesystem in the /etc/fstab file.
/etc/fstab: /dev/sda1 /tmp ext4 rw,exec 0 0
Keep in mind that changing the mount options for the TMPDIR filesystem may have security implications, so you should be careful when making these changes.