Yes, that's right. The whole sentence can be rephrased "Although he is popular (or Despite being popular), the President hasn't always managed to have his own way."

Yes, the "he" refers to the president. "As popular as he is..." is the same as saying "Even though he is so popular..." So the whole sentence could be simplified to: "The president is extremely popular, but he hasn't always gotten what he wanted." Hope that helps :)