I haven't looked at the install procedure for wordpress, but it should be ok if you go into filemanager, create new folder, name it anything, maybe 'test'. Upload wordpress to the 'test' folder, create a database for it then follow the wordpress install procedure. You can then get to the wordpress folder by simply typing your url and adding /test to the end. Once you've transferred all your content from one site to the other, you will need to 'swap'. Your database settings will need to be changed, probably from within config.php these will mainly refer to the url and path to directory. If you're still unsure, just edit and i'll look a bit deeper into the wordpress install procedure for you.
Thank JK for the addendum, haven't used wordpress so wasn't so sure about the final touches. However i would add that it's not necessary to downlod the files.
After it's all set up and updated ready to go, go into the directory of your old site which is probably public_html. Use the select all function then un-tick the folder that holds the new site, in my example 'test'. Select the delete button to get rid of all the old files, leaving only the new wordpress folder.
Now, just in case there's a hiccup, open the wordpress folder, use select all, then use the 'copy' button. Copy them all out of the new folder, so if the folder name is 'test' (public_html/test) delete the word 'test' from the url offered to you. Now your site will be in the public_html folder AND still in the original folder. So if anything goes wrong you've still got the site in 'test'. Once you've set the correct url's and you're happy that it's all functioning you can then delete the 'test' folder.