You are getting at two things. One is to pay early for interest the other is to pay early for taxes at the end of the year. it is possible that this varies from bank to bank, but here are my experiences.
Paying two weeks early doesn't really help you because the bank posts assuming that you are paying your cycle. I am not sure the best way to explain this, but the principle reduction is assumed to occur on the the 1st of the month whether you pay 2 weeks early or 2 weeks late (on the 15th).
The other question about paying your January 1st bill on December 31st (or earlier) can help taxes because the bank will report that you paid your interest on the day your payment posts. So if you paid all your January 2011 statement on January 5th, 2011, and all your other statements on time, then make your January 2012 payment on December 28th, 2011, the bank will essentially report that you paid 13 months worth of interest in 2011, so you will get an added tax benefit. Of course if you do not pay the January 2013 payment early, then your 2012 taxes will only have 11 months of interest.... more