when I needed to upgrade my site to the latest Sitefinity 3 version, it was running on shared hosting too. I was unable to perform the upgrade in place because of restrictions on permissions on the live server.
I needed to download the site to my local machine, apply the upgrade locally, then migrate that back to the live site.
If you are on shared hosting you'll likely have to do the same. It's recommended that you do this anyway because that way if you encounter any issues, your live site doesn't go down, and is only updated after you know everything works.
Plus, if you download your site, update it locally to 3.7 SP4, you can easily move on to migration from there, since it's recommended you upgrade from a local copy anyway.
I hope this is helpful!