Yes, it will work if you have Cubase Elements 8 or higher. It says so at the bottom of the page.
The reason why it says that it's only included with Artist and Pro at the top is because Elements 8 originally didn't ship with the Acoustic Agent drum kit, which is required for these styles to work. They later made this kit available for Cubase Elements users to download, and Cubase Elements 9 and later come with it installed by default.
Note that Style expansions like Jazz Essentials don't add just new rhythms, they also come with Groove Agent mixer presets that can drastically change the sound of the drum kit.
Every time someone says "it must be easy to add/fix", a programmer dies.
Cubase Pro and Wavelab Pro (latest), Windows 7 Ultimate 64-bit, MSI Z87-G45 GAMING, i5 4690k, GeForce GTX 760, almost every Steinberg plugin and expansion, Trilian, Komplete 10, etc etc etc etc