You need to install the free Creative Cloud desktop application to sync any fonts from Typekit; that software is how Typekit transfers the font files to your computer.
There is a link to the download from the first section of the font sync tutorial:
Once Korolev is listed as Synced in the Creative Cloud desktop application, the fonts will be available from the font menu of your desktop programs.
If you aren't able to install the Creative Cloud desktop application on this computer, then you won't be able to sync fonts from Typekit. Instead, you could look at purchasing a perpetual desktop license for Korolev from the foundry that owns it, Device Fonts. That desktop license would allow you to download and install the fonts locally on the computer.
I hope that this helps; let us know if you have any other questions.