Time is one factor that often becomes a problem for rich people because they are too busy accumulating wealth. Another problem that rich people fear most is wealth itself, they are very afraid of losing their wealth because they are used to living in luxury and are used to seeing balances with long rows of "0". Another problem that comes to rich people may be the loss of their privacy due to becoming famous for their wealth. In essence, every human being has their own problems, so even rich people definitely have problems, they don't have problems when they leave this world.
Times is generally anyone's problem in that context because time does not discriminate against anyone and always works simultaneously, whether knowingly or unknowingly. I know your argument will be that the rich people are mostly more conscious of the time than the poor people but it is not always the case because time is generally a personal phenomenal that every individual experiences and values differently.
I like to use things that are more practical Trust issues, loneliness, social pressures, security concerns and pressure to always try to maintain their wealth. Most rich people have issues with entrusting their valuables with people because in most places, people who are entrust with direct access to wealths as they have been scammed and duped repeatedly.
They also have issues with being lonely as many wealthy people require security and privacy. I like to take both their privacy and security concerns as the result of their loneliness since they can't move freely on the streets as a poor person would. They also become lonely and isolated sometimes because of the social pressure that's mounted on them by the people. The rich people always call for attention of the other persons whether they are deliberate or not. Due to that, they become a little bit more restricted and protected to avoid being exposed to bad eggs.