No idea about how to do it in picassa but if I wanted to do what you want to do, I'd use
ImageMagick. Specifically
http://www.imagemagick.org/Usage/annotating/
It's probably already installed on your machine, if not check the repos and it'll be there.
Though I have the feeling there ought to be a much better way to do that but can't just figure one out right now.
Code:
for i in *.jpg;do
year=$(ls --full-time ${i} | awk '{print $6}' | cut -d - -f 1)
see_linked_examples_for_a_convert_command_to_use_here_you_can_use_the_variable_year_in_the_text
done
I have a feeling there should be a much better way to get the year a file was modified than the example above.