1. Take the high-speed train from Shenyang North Station to Beijing Station, and then take a taxi to the Palace Museum. It takes about 3 hours and 48 minutes and costs about 360 yuan.

2. Take the train from Shenyang North Station to Beijing Fengtai Station, and then take a taxi to the Palace Museum. It takes about 7 hours and 3 minutes, and the cost is about 103 yuan.