To avoid that in future before filling the partition, it’s a good idea to install and run periodically: purge-old-kernels. Installation and example of use are:
apt-get install bikeshed
# keep three old kernels:
purge-old-kernels --keep 3
# if you want to put that in the crontab use that command
purge-old-kernels --keep 3 -qy
If you’re a Grub user don’t forget to run:
Personally I have a nightmare with that problem and Ubuntu, especially with version 12.04 which is installed in a lot of servers that I manage. I repeated the previous process a lot of times and in the end, I decided to document it because I always have to go to Google and find the proper steps to solve that problem.