In live session I don't think it is possible to install a newer kernel because you have to reboot.
To install a newer kernel on Zorin you can use e.g. cappelican mainline tool and follow this guide:
Probably you will also need newer mesa drivers. You can get them by adding the ppa kisak mesa fresh (or maybe it also works with ppa kisak mesa stable).
(you can still use the fresh ppa in Zorin - Aravisian has told me recently)