What seems to have happened:
Your / was full, then Ubuntu created a new partition, in RAM memory, to use temporarily.
Now, this 1MB partition is not big enough for the job, either.
What we can do:
1) increase the size of this partition just to do the upgrade
2) actually delete enough files in the HD that this partition is no longer needed.
To do 1:
open a terminal and run
sudo umount /tmp
sudo mount -t tmpfs -o size=1048576,mode=1777 overflow /tmp
This should give you an 1MB partition (just like the one you had =P).
Now, to increase the size, you increase the size in that line, so that, with size=10485760, you'd get 10 MB.
Your goal is to find a number that is enough for the job, but leaves enough ram too
Comments on 1
You may want to try sudo umount -l /tmp, if you get some variation of "the file system is busy and cannot be unmounted"
Another possible solution to "the file system s busy(...)" is to do fuser -m /tmp to find pids (process numbers) that are using /tmp, then ps -elf <pids>, stop or kill processes
You may want to try sudo mount -t tmpfs -o size=1MB,mode=1777 overflow /tmp or even sudo mount -t tmpfs -o size=1G,mode=1777 overflow /tmp (for 1 megabyte or 1 gigabyte, respectively) - that is, units are available so that you dont have to type a huge number
To do 2:
Open a terminal and run sudo umount /tmp or, if that fails, sudo umount -l /tmp.
Then clean up!
Delete files in /tmp (now /tmp is the thing actually in your HD, rather than a virtual ram disk), uninstall unused packages, delete files in your home folder and so on.