There is no specific time to pay the zakat. You should determine your own period.
The annual paying tradition comes from the earlier times in which most of the people were earning their subsistence annually. Now we are living the times in which most of the people are earning their income monthly. So it is much more rational to give the zakat monthly. But anyway, it is up to you and your own conditions.
Zakat has nothing in particular to do with Ramadan.