I followed the instructions here to upgrade the kernel on my SUSE 8.1 install. The instructions are written assuming LILO is the boot loader so I'm not sure if the part about editing LILO would apply to grub.

I think I have successfully updated the kernel but grub isn't showing an option to boot into the new one. Here are the lines I added to grub:

image = /boot/bzImage-2.4.22
root = /dev/hdb1
label = new_kernel
read-only

Here are the lines from /boot/grub/menu.lst that apply:

title /dev/hdb1_linux
kernel (hd1,0)/boot/vmlinuz root=/dev/hdb1 hdc=ide-scsi vga=791
initrd (hd1,0)/boot/initrd

If this is correct, how to I append these lines to grub so it will give me the option to boot into the updated kernel?

I tried adding the image, root, label, read-only lines after the initrd line but the old kernel wouldn't boot that way.

Those lines are for LILO.
Replace vmlinuz with bzImage-2.4.22 and it should boot, I guess.

I already tried that. I've tried several things already so I'm not sure what error I got with that one. It was either "File not found" or "Kernel panic: unable to mount root fs."

Are you sure your root partition is /dev/hdb1? Isn't that your boot partition?