What is the cheapest way to go to our hotel Lao Zhang Inn Lao Zhang's Inn - Long Qing Xia ?

To Lao Zhang Inn, you are advised to take shuttle bus line 1 to Dabeiyao (Guomao Bridge) and then walk to Gumao metro station to take Beijing subway line 1 to get off at Gucheng Station. Then its exit A is around 1,000 meters to your destination!

To the wall (I presume you refer to Badaling Section), take subway line 1 from Gucheng Station to Fuxingmen, where transfer to line2 to Xizhimen. Then walk to Beijing North Train Station to take a train to reach.

Then from the wall to Ming Tombs, you are advised to use bus no. 879, which will take you to Dingling and Changling!