1) The Sefardic custom is to recite Selichot (additional supplications before the morning services) throughout the month of Elul, whereas the Ashkenazim only commence on the Sunday before the new Year. If the New Year falls on a Monday or Tuesday, one starts the Selichos the week before, so that there will be at least four full days of Selichos before the New Year.
2) There are two reasons for having at least four days of selichos: 1) Since there is a custom to fast during the ten days of repentance and there are four days when one cannot fast (both days of the New Year, Shabbos and the eve of Yom Kipur) so we make up for those days by reciting selichos for four extra days. 2) About the additional offering on the Ney Year the verse says “And you shall make an offering” (instead of the usual “And you shall bring an offering close”), to teach us that each one of us should make as if he is offering himself, and in the same way as an offering needs checking for blemishes for four days before being offered up, so too a person should check himself for sin for four days before the New Year. (Mishne Berurah 581)