Install Magento 1.5 via Plesk, then go to /downloader.
Use the key http://connect20.magentocommerce.com/community/Mage_All_Latest to update files.
Magento upgrades the DB automatically on next load.
Or unzip 1.6 files over 1.5 using Plesk/FTP, then visit the site to trigger DB upgrade.
No SSH needed.