In the Name of Allah, The Most Gracious, Most Merciful
Paying Zakat is a financial obligation. It would be better to pay it at the beginning of Ramadan, because if it is postponed to Eid, the one who receives it will not have the time to use the money for his needs (like buying clothes to his children).
There is another suggestion: one can pay his Zakat in Shawwal, by which he pays in advance the Zakat of the coming year. If you know a mother whose child is in need of a surgery, you can pay for it.
Getting your Zakat ready at the beginning of the year gives you a chance to pay for those who deserve it. Paying in advance makes your money ready to be paid for cases you find worthy of your Zakat.