How can a Muslim who excommunicated his brother without right repent, and does he become an infidel by doing so, especially if the one he excommunicated does not pray?

Unjustly declaring a Muslim an infidel is a sin. The meaning of the Hadith: "Whoever says to his brother, 'O disbeliever,' then it has surely returned upon one of them," is to emphasize the gravity of the act, not that the speaker himself becomes a disbeliever. Repentance from declaring a Muslim an infidel necessitates seeking absolution from the rights of other people, similar to all sins related to them. The rights of Allah are fulfilled through regret, cessation, and resolve. However, the rights of other people (life, property, honor, and hearts) require seeking forgiveness and absolution from the wronged party, or accumulating good deeds for the deceased or absent. Whoever has been subjected to emotional harm must seek absolution from the one who harmed them. As for one who abandons prayer out of negligence and laziness, he is not considered a disbeliever by the majority of scholars.