Making full course backups to restore to a 4.1 is the path of least resistance, but ... there is always one of those 'buts'! 
In the 4.1, best if you have all the additional plugins that were installed in the 3.1 installed in the 4.1. Theme, course format, question types, etc., etc.
If you have enough space on old server could setup automated full user backups to a designated directory ... outside of moodledata/filedir/ so you can see them and manipulate their location.
If all those complete successfully and you have the 4.1 prep'd, then it's a matter of xfering the course backups onto the new server in a file system repo where they can be restored.
On the new server, one needs a cPanel (or any panel) that provides 2 'advanced' things - Terminal and Git. Yes, Terminal is command line. Moodle server admin is best done via command line where it makes sense to use CLI ... and me thinks this is one of those times!
Git ... I'd install the new 4.1 instance via git versioning ... that will make your admin life much easier when it comes to updating/upgrading your Moodle.
My 2 cents, of course!
'SoS', Ken