Yes, you can move your changes to a child theme - first make a complete back up of all your files.
Then create a child theme per the instructions here - http://codex.wordpress.org/Child_Themes
Then move a copy of any .php file (EXCEPT functions.php and style.css) you have modified into the child theme folder.
For style.css, you will need to compare your file to the unmodified version - there are file comparison tools that can help you with this. Put only the changes to CSS in the child theme style.css file.
If you've made changes to the functions.php file, you'll need to read the instructions in the Codex as to how to put those in a child theme.
Once you've done the above, activate your child theme. Note that theme settings and options do not transfer to a child theme (like any other new theme) so you will have to redo those.
Once that's done replace/update the parent theme.
Again, make frequent archive copies of your files as you are working on this - so that if you mess up, you can easily go back a step.