I doesn’t matter if you cross compiled it or not, this is how you run it:
Use e.g. scp to copy the new zImage to the Jetson with a new file name, e.g. /boot/zImage-my01
Then edit /boot/extlinux/extlinux.conf. Copy the existing menu entry and change the name and the kernel image file name.
Connect the serial cable to Jetson if you don’t have it already. In practice do always do need the serial cable when modifying the bootloader or the kernel.
When Jetson is booting up you can see bootloader showing the kernels listed in extlinux.conf. Select your new kernel and the bootloader will boot that. If your kernel fails to boot, just hit the reset button and let the bootloader boot the default kernel.
After you have verified that your kernel works fine, you can change the default kernel from extlinux.conf and the bootloader will then boot that by default.