The problem with anyone trying to give information other than that in a tutorial/FAQ is that there are just too many variables that we have no knowledge of AND we might leave out a step that is vital to your success. I'm guessing that you're not familiar with FTP, file editing, and the like. Not knocking you, just most folks who are won't use a shortcut to load and setup files.
That said, here's some things to consider.
Your host might be willing to add a redirect from the root to the zencart folder. That would redirect anyone typing in yourwebsite.com to yourwebsite.com/zencat but their browser would look like they are still at yourwebsite.com. This can cause problems down the road with some mods, but could get you going temporarily.
If you have done little customization (mods, templates, etc.), you might want to do a manual install into the root directory and import your current database. Of course, you don't want to jump into any of this without a backup of BOTH the database and the site.
Does your host provide you with cpanel? If so, you have some options available to you but should not be entered into without proper planning.