How to say incoming president (国家主席)?
即将就任 is the term used to describe incoming. Example 即将就任的国家主席. Normally, the Chinese word for president is 总统 and 主席 is usually used to describe chairman or chairperson. I note that China has changed the title of 国家主席 from Chairman to President from 1983 onwards.

The appointment 国家主席 is rather symbolic in nature as the real power lies with other political appointments (中国共产党中央委员会总书记 and 中国共产党中央军事委员会主席) that usually precedes it. As of current, 习近平 has yet been appointed 国家主席, but he already held the two most important appointments.