I have read somewhere that the Prophet (sws) disliked that anyone should stand up for him in respect. What should we do in this age where showing respect by standing up for the other person has become a vital part of good manners?

The Prophet’s dislike for standing up in respect for him by his followers was an expression of humility on his part, and of course does not mean that standing up in respect for someone is a bad thing. The Prophet (sws) did not like it for himself, and many of our own elders do not like it either.

If an elderly and respectful person for whom you stand up stops you from doing so, then your compliance in this regard is out of sheer respect for his wishes, otherwise he certainly deserves this since standing up for him is just meant to show respect to him.