Is it Sunnah not to stand up and shake one's hand if I was sitting? Or do I have to stand up and shake hands? What if the man who came to me to shake hands was older than me?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Getting up for a visiting person who has any significant quality such as knowledge, piousness, rulership or aging etc., is desirable provided the standing should be with the notion of respecting him not for glorification or showing off. The evidence for this is that the Prophet (Sallallahu Alaihi wa Sallam) said to Ansar (Radiya Allahu Anhum) "Get up for your chief" or he said: 'get up for the best amongst you" . Meaning S`ad bin Muadh (Radiya Allahu Anhu) [ al-Bukhari ].
Imam al-Nawawi said: 'we believe that getting up to honor a respectable person or a knowledgeable personality, or a noble person, or a pious one, or an aged relative or the one who has high authority is likeable provided it should only for the purpose of honoring and showing respect for him not for glorification or showing off. Our ancestors and their successors used to do so. [al-Azkar].
Allah knows best.