Well you'll have made backups, of course.
Open in 2013 (you may get asked what you want to do with 2003 file)
Make a minor change to one cell, then Undo or manually put it right.
Click Save and see what it says. Click Save As and see too (note the dropdown box for old versions. It will allow you to keep as old version if you want).
Close Excel and re-open with same file. Check it all seems to work, formatting and printing is OK).
When you are happy, embark on lengthier editing sessions.