A nationwide Japan Rail Pass does not cover a Shinkansen "Nozomi" or "Mizuho" ride.
According to current timetables,
if you take Haruka 26 / 28 / 30
(departing at 16:16 / 16:46 / 17:16, arriving to Shin-Osaka at 17:07 / 17:38 / 18:07),
you can catch Shinkansen Sakura 569 / Hikari 477 / Sakura 571
(departing at 17:20 / 18:05 / 18:20 at Track 21, arriving to Himeji at 17:50 / 18:42 / 18:50).
In Shin-Osaka Station, you can go up via elevator or escalator to the concourse, pass through a transfer gate to Shinkansen and go up via elevator or escalator to the Track 21 & 22 platform.
In Himeji Station, you can go down via elevator to the Shinkansen concourse, pass through a transfer gate, go down via elevator to the non-Shinkansen concourse and pass through Central Gate for North Exit and South Exit.
Tourist Information Center "Himeji Kanko Navi Port" is near North Exit of Himeji Station.
http://www.himeji-kanko.jp/en/spot/kanko.htmlBy the way, will you stay overnight in Himeji City?
Even if your plane arrives (almost) as scheduled around 15:00, Himeji Castle and Koko-en garden will get closed by the time you reach them or soon after that.
The exterior of the renovated Main Keep of Himeji Castle has already appeared, but visitors cannot get inside the Main Keep until the ground opening scheduled on 27 March 2015.