Precisely 13 months.
He came into posession of this chunk in the beginning of Episode 18 (Season 1): "Injustice for All".
The episode begins with Lex Luthor standing over Superman, brandishing the same kryptonite brick... or so it seems. It turns out that it was J'onn J'onzz, impersonating Superman in order to retrieve the kryptonite (and some information).
If you watch carefully, you'll notice Green Lantern take the kryptonite from Lex, using his ring beam - and Batman snatches it out of the beam, and pops it into his belt.
Injustice for all was aired September, 2002 - Tabula Rasa (your episode) in October, 2003. Precisely 13 months.
A footnote, regarding your final question - does it have any effect?
This question is especially pertinent, since it comes into Batman's posession in the context of the fatal kryptonite poisoning that affected Luthor.
However, it was established that the pouch on his utility belt was lead-lined, completely blocking all radiation from leaking.