Is it permissible to buy new clothes for the eldest son and then have the younger son wear them, given a one-year age difference between them, and is this considered a lack of fairness among the children?

Justice among children is, according to the more preponderant opinion, obligatory. Observing custom is the principle regarding clothing. So, if the older child needs new clothes due to the necessity of his appearance among people, and the younger one is content with old clothes, this is not considered injustice. It is permissible to favor some children for a need, such as old age, illness, a large family, or engagement in seeking knowledge, or to withhold a gift from a transgressor or an innovator. However, in the questioner's case, it is unlikely for custom to change between the two sons due to the age difference, and therefore, it is not just to always allocate new clothes to the older child.