I am going to assume that your society is not very different from our society in all other aspects. If it is the case some psychological and social differences will be:
1. Concepts of age and ageing
We use visual appearances such as wrinkles, saggy skin, dull skin tone, grey or white hair, loss of muscles, etc. as indicators of age because this is how ageing happens in our species. In your society, these visual characteristics will indicate a phase rather than an age since all people go through these phases.
Unlike our society, the concept of age will be disassociated from appearance. Therefore, there will be no inner conflicts caused by young appearance and multitudes of experiences. The young phase or old phase are more likely to be associated with the current level of capabilities and social obligations. For example, people in their 2nd cycle / young phase might be expected to get married and have children. While people in their 3d cycle / old phase are expected to reflect on their lives so far and prepare to make adjustments as soon as they become more physically capable due to the 'de-ageing'.
Age will be defined based on the number of completed cycles (ageing-'de-ageing') and current phase instead of decades that we use. Years lived will be used when a more precise age assessment is required.
Old persons will be people who are in their last cycle (or what is considered to be 'the last cycle') no matter the phase.
If your culture values seniority then your society will come up with ways to indicate one's age using various means: Clothing, manners, addresses, social positions, etc. Clothing is probably one of the most suitable mediums to serve as a symbol of the age. You can check various cultures and history of fashion to see how clothing has been used historically to symbolise one's age and status.
2. Determination of age seniority
If your culture employs some highly visible symbols of age, the concept of seniority might not be different from ours. Except instead of relying on the physical appearance of one's body people will rely on those symbols. However, if there are no easy-to-distinguish indicators of age/experience your society will have to come up with some other means of determining seniority by age. For example, people may use some conversational clues to find out someone's age (this is a very common approach in cultures where women do not disclose their age).
Some modern cultures do not regard age seniority as important or even practise ageism. Your society can be one of these. Seniority may be decided based on something unrelated to age, e.g. merits or birth status.
3. 'Suitable ages'
Our societies have notions of suitable ages for doing something, for example, marriage, child-rearing, schooling, etc. These notions are based on the development of our bodies and social expectations/norms. Instead of 'suitable age' your people will have a suitable phase or a suitable cycle (1 young phase + 1 old phase). The 'suitability' of those phases or cycles will be associated with body condition (old or young) and social norms (see 1. paragraph 2).
It is also reasonable to assume that your society will develop 'cyclic' obligations and expectations. For example, people may be expected to have children every time they reach the physical peak of the 'young' phase. Or they may be expected to retire for 10-20 years when they reach the peak of the 'old' phase (i.e. they are physically the oldest). This is up to you to decide and most of these expectations are arbitrary unless they deal with physical prowess or phases result in a noticeable decline in cognitive abilities.
Since your people's lives go through cycles the concepts of time will be affected accordingly. It is not hard to imagine that instead of the linear time that we use your society will have a notion of cyclic time or spiral time. Cycles and cyclicity will probably dominate sciences and everyday life and give life to many alien to us philosophies and traditions. It is impossible to say what exactly people could think of, so you, as an author, have a lot of creative freedom here and can come up with a multitude of ideas based on cycles.
I started this answer by saying that I am going to assume that your society is not very different from our society. However, if you want to make your society realistic you should consider that just the 4 things I mentioned will lead to cultural developments very much unlike those in our world. The single fact that there is no reliable way to determine the age of an adult would change the culture immensely.
After writing this wall of text... A simple answer to your question is: A natural-born person of your world will think that it is completely normal to become young after being old several times. There will be absolutely no psychological problems since this is a part of a normal human life cycle in your world.