How to transfer data from mysql 5.5.24 to 5.1.36
A backup is just a boat anchor until you verify that you can reload from it.  So, give it a try.  Set up a 5.5 version (if you can find such an old version) and try the dump.

It may work as is.

How are you doing the backup?  mysqldump?  If so...

At worst, you might need to edit the dump file to fix some old syntax.  (I don't remember any from 5.1 to 5.5.  Before that, there was TYPE=InnoDB changing to ENGINE=InnoDB.)

If you are copying a directory tree, then "files are files"; there is the backup is just a filesystem; it does not care what the bits mean.

So, I am puzzled as to why you talk about "5.1", yet you talk about "/data".