You can use
stat to query the mode on your backup file, and
chmod to update your working files. Do this in a loop, with a little string manipulation, and you have a solution. Something like this should work to copy the permissions:
# path to the top level of the backup
# path to the directory of files that you want to restore
# permissions on.
find "$backup_dir" |
while read backup_file
chmod --reference="$backup_file" "$new_file"
Note that this won't work well if your filenames contain the newline character (but thats's pretty rare). Also, it's untested, so you may want to do a "dry-run" by changing 'chmod' to 'echo chmod', so it just prints what it would do.