I’m afraid it’s not possible to install Discourse on to a web hosting package as you describe.
Discourse requires ssh access because unlike phpBB it’s an application, instead of a collection of php files.
Discourse will install docker, and then within docker it will create an image which includes everything it requires to run.
Take a look at the standard install guide. A $5 VPS at DigitalOcean is ample to get started.