Finally, I managed to get to my goal of running newest kernel on my cubieboard:
root@cubieboard2:~# uname -a
Linux cubieboard2 4.4.0-rc6-sunxi #3 SMP Mon Dec 28 11:38:20 CET 2015 armv7l GNU/Linux
So, how did I get there, you might ask? First, I noticed that my kernel on laptop was reporting bad geometry for sdcard so I re-created partition table to fix it.
I also wrote u-boot again to card using
dd if=u-boot-sunxi-with-spl.bin of=/dev/sdX bs=1024 seek=8
Have in mind also that I already re-created ext4 filesystem previously on this card. My conclusion is that cubieboard2 image on Igor's site is somehow corrupted, and that one of these steps (partitions, mkfs.ext4 and/or u-boot) fixed it.
In next step, I did apt-get install linux-firmware-image-dev-sunxi linux-headers-dev-sunxi linux-image-dev-sunxi linux-u-boot-cubieboard2-dev
which installed latest versions of linux and u-boot.
I'm happy 4.4 kernel user now, and want to thank Igor for his hard work on armbian. Hopefully, this post might help somebody else in future.