Your shared hosting service doesn't need to support Let's Encrypt, it just needs to allow you to upload a certificate. You can use https://gethttpsforfree.com/ to generate it.
The renewal is the same, follow the steps and you get a new cert. Keep the Account key and the CSR, so it's just a matter of copy-pasting. If they charge an installation fee again, you're probably better off paying for cert that lasts longer (not from Let's Encrypt) - you can get one that lasts three years for $15. Or just switch hosting providers :)