Is there any transport from Kashgar airport to Tashkurgan beside by international bus?

Well, as I know, there are only two buses from the Bus Station in the opposite of the Railway Station to your destination, which leave at 10:00 and 11:00, take CNY55.
By the way, you can take bus 2 to Jiaotongchu, and then walk south to Minmao Bus Stop to take bus 28 to the railway station.
If you can’t catch the bus, you may charter a local car to there, which may take around CNY500, and you may share the fare with others.