Japan Rail Pass

I am going to Japan at end of January. There seems to be quite number of passes available.
I would be appreciate if someone could help me on which is the best pass to purchase before my arrival. Below is my itinerary
Day 1.
From Kansai airport to Kyoto.
Day 2, 3
In Kyoto
Day 4
Kyoto to Osaka
Day 5
Osaka to Nara. Back to Osaka
Day 6
Osaka. Going to Universal studios
Day 7
In Osaka
Day 8
Osaka to Himeiji. Back to Osaka.
Day 9
Osaka to Kansai Airport.

Thank you.

No rail pass will pay off for you as you're travelling short distances. But you could consider an Icoca and Haruka pass - this gives you a round trip from Kansai Airport (to be used within two weeks) and an Icoca prepaid card which can be used for trains and subsways everywhere you are going, which comes preloaded with 2000 yen. You can load more yen on this card as you need and refund the remaining amount at the airport before you leave.

I suggest you stay near OSAKA station. Also there are several subway lines, the Hankyu and Hanshin private railways (UMEDA station).
From the airport (KIX) there is a special combination ticket using the Nankai railway and Osaka subways to and from your hotel in Osaka. It was 890 Yen before the tax increase. Now ?
Here are the various places you want to go from Osaka station area.
To Kyoto JR station using JR Rapid train. 560Yen, 30 mins.
To Arashiyama or Karasuma stations in Kyoto using the Hankyu Ry. 400 Yen, 55 mins.
To Himeji:
Using JR Shinkansen. 3220 Yen, 30 mins.
Using JR Rapid train. 1490 Yen, 70 mins.
Using Hankyu or Hanshin Rys. 1280 Yen, 90 mins.

From the airport (KIX) there is a special combination ticket using the Nankai railway and Osaka subways to and from your hotel in Osaka. It was 890 Yen before the tax increase. Now ?

I don't recomment using those. The trip using the Icoca & Haruka package is only marginally more expensive (1030 yen) and much less hassle (no need to make a transfer and take the potentially crowded subway).

(Yes, despite its name, the Icoca & Haruka is also valid on the Kansai Airport Rapid Service.)

Use the ICOCA & HARUKA round trip.
use ICOCA while in kyoto,
use Kansai Wide pass for 4 days from Kyoto to Himeji.
May not be cheapest, but save time buying individual tickets and allow to travel on any trains eg shinkansen.

Jensen's suggestions are sound. With the JR Kansai Wide pass (which you don't have to book ahead, you can buy it at the ticket office at Kansai Airport) will let you ride all express trains in the area plus any Shinkansen (including the Hikari and Mizuho) between Osaka and Okayama. Not 100% it ends up cheaper than riding the normal express train to Himeji but riding the Shinkansen is an attraction in itself so I'd definitely recommend it. Hell, get the pass and do Okayama on the same day you're doing Himeji. The Korakuen is lovely any time of the year and walking up to the reconstructed castle is nice (though I wouldn't bother actually going inside). Won't take you long either.