Is FIDE president Kirsan Ilyumzhinov a good chess player? What's his Elo rating? On this page it says he is not rated: http://ratings.fide.com/card.phtml?event=34123811

I could only find one game featuring Ilyumzhinov, in which he played a 2300 Elo player and lost: http://www.chessgames.com/perl/chessgame?gid=1600034

That's not very impressive for someone who owns FIDE and who boasts of having played a few games with aliens...
According to the source provided by @Lynob, Kirsan Ilyumzhinov had a rating of 2300 and was local champion of Kalmykia at the age of 14 (see also here). The game you mention against Alexander Zakharov took place in 1978, when Ilyumzhinov was about 16 years old.

However, and as you say, Kirsan Ilyumzhinov is unrated at the moment according to FIDE. But according to the same source he has played 9 rated games (not enough to get a rating). Making some research I found he played these games in the Members Appreciation Blitz Tournament in Skokie (USA) in 2012 with a result of +4 =1 -4.

Appart from that, I think he mostly plays friendly games. If you search "Kirsan Ilyumzhinov plays chess" in Google you'll find some images of him playing with different people, such as Gaddafi, Steven Seagal, the Minister of Sports Affairs of Oman or a Paraguayan student.