No way around ... a lot of work ahead of ya! On shared hosting is problematic. Did you begin with a full site backup (moodle code + moodledata/filedir/ + DB dump? Hope so ... with those one could download and then begin working on a plan.
check out PHP/MySQL/MariaDB requirements
plan ... install virtual box on local computer ... then intall linux version as close to hosting as you can get ... but make it minimal ... just ssh to begin. Build a box with appropriate versions of PHP/MySQL/MariaDB to be able to restore the backup.
You will have to edit config.php and probably do a search and replace of the SQL dump for https://fqdn/ to http://private_ip_of_VB/
Once you have that 2.x.x running, login as admin and go to Site Admin, Server, Environment check - update the component.
Use the pick list to help plan ... one can do some hops with php 5.3 and older MySQL/MariaDB, but before a hop that requires higher versions, one has to update the PHP/DB server ... not to latest/greatest but what the next hop in Moodle can run under.
Make the hop upgrade ... check the site - do a backup of code + DB to be able to restore in case the next hop fails for some reason.
You will also probably run into having to upgrade themes and addons/plugins to the moodle along the way. Some themes no longer compat with higher versions of Moodle so you'll need to edit config.php to force a compat theme so you can then remove the theme via moodle admin UX.
You have to make sure that where you land on that VB instance, it can be restored to the shared hosting instance ... usually not a problem downloading backups of a Moodle, but uploading and restoring could be ... depending upon the level of shared hosting package ... the cheapo 5.95 a month is a red flag.
Think you are in for a very long weekend! :|
Best of luck!