The trip, under way as The Economist went to press, began inSaudi Arabia(whose king, Salman bin Abdul Aziz, is pictured with Mr Xi).

副词完全可以修饰主语或名词的！你的句子中的under way 是介词短语，在此用作非限制性定语，修饰名词 the trip。under way 后面的 as The Economist went to press 为时间状语，意思是“当这期《经济学人》准备付印之时”。全句大意为：当这期《经济学人》准备付印时，这次从沙特阿拉伯启程的访问仍在进行中。