There is a custom observed by some people to fast on Ereb Rosh Hashanah. Although there are those who fast every month on the day before Rosh Hodesh a fast called "Yom Kippur Katan" the custom to fast on Ereb Rosh Hashanah is mentioned by the Shulhan Aruch, and is thus to be treated with special gravity.
The source for this custom is a passage in the Midrash, cited by the Tur, which draws an analogy to a country whose citizens owed an enormous sum of money in back taxes which they could not afford to pay. They decided to send a small delegation to the king, who agreed to cancel one-third of the debt. But as the day approached when the payment was due, the people realized that they still had nowhere near the amount. They therefore sent a much larger delegation to plead their case before the king. Once again, the king relented and waived another third. The day came, and the people still found themselves unable to pay. The entire citizenry all the men, women and children came out to greet the king with tears and pleas, and the king accepted their pleas and waived the entire debt. Similarly, the Midrash comments, on Ereb Rosh Hashanah, as we realize that we are unable to "pay our debt" to the Almighty, select individuals volunteer to fast and plead to G-d, who waives one-third of our "debt." During the Aseret Yemeh Teshuba between Rosh Hashanah and Yom Kippur, as we get even closer, more people come before G-d in fasting to ask for a reprieve, and another third is cancelled. Finally, on Yom Kippur, we all fast and G-d grants us complete forgiveness.
It is therefore appropriate for anyone who can including youngsters above the age of Bar Misva to observe this fast on Ereb Rosh Hashanah to be part of this initial "delegation" which comes before G-d to ask for forgiveness. Needless to say, those who are frail or elderly, and those who have trouble fasting, should not observe this fast, as it is not strictly required.
One might ask how it is possible that the voluntary fast of Ereb Rosh Hashanah can accomplish just as much as the obligatory fast of Yom Kippur. According to the Midrash, both these fasts are capable of atoning for one-third of our sins. Is it conceivable that the fast of Ereb Rosh Hashanah is as significant and effective as the fast of Yom Kippur?
Rabbi Hillel Haber explained that the first "reduction" is always the easiest. If a person wants to purchase a piece of merchandise and sets out to negotiate a price reduction, the seller might show some flexibility at first and agree to reduce the price by one-third. But afterward, he will be much less likely to agree to a further reduction. Certainly, he will not likely be prepared to offer the merchandise free of charge. Accordingly, the customary fast of Ereb Rosh Hashanah accomplishes the relatively "easy" task of securing a reduction. But the fast of Yom Kippur accomplishes something far greater allowing us to earn complete atonement "for free."
It should be noted that those who normally fast on Ereb Rosh Hodesh will fast the day before Ereb Rosh Hodesh if the Molad (first appearance of the new moon) occurs before that day. The Yom Kippur Katan fast must be observed before the Molad, and it thus occasionally becomes necessary to fast the day before Ereb Rosh Hodesh. When it comes to the fast of Ereb Rosh Hashanah, however, the time of the Molad is immaterial. Even if the Molad occurs before Ereb Rosh Hashanah, nevertheless, the fast is observed on the day of Ereb Rosh Hashanah.
Summary: There is a custom to fast on Ereb Rosh Hashanah, and those who are able to observe this custom without too much difficulty should do so.
