Can't say that I have done something like that myself, so no guarantees, and you'll most definitely want to make a
full backup (filesystem
and database) before you try anything.
Your best bet probably really is to update straight to the latest version with the
usual update process – as for "smoothly", no idea. I remember some people updating from pretty old versions back when 2.0 came out, but I don't recall anyone going from 1.3.1 to 2.1.x specifically.
I would also suggest to wait and see if someone else has a different idea, but I don't see any. You would need to update even if your hoster still had 5.x lying around, 1.3.1 is 10 years old and full of security issues.
Make sure you go with PHP 7.0.x, not necessarily 7.1.x and definitely not 7.2.x. Those are not fully supported yet. Good luck.
YL