(Edit spelling)Batman cannot beat Superman in a fair fight. Kryptonite ring or no, a flash of heat vision at a distance would vaporize everything it touches. Or a boulder thrown with super-strength and speed would pulverize anything it hit. So a fair fight is out.
Fortunately Batman doesn't always fight fair. Unlike Lex Luthor, Batman would not underestimate Superman.
Using his knowledge of how Superman does things, Batman could cause an explosion causing a large fire. As Superman arrives, Batman can saturate the air with Kryptonite dust, knowing that Superman will inhale a large amount of air to blow out the fire. Superman breathes in the kryptonite and is weakened enough for Batman to defeat him.
Since Batman knows Superman is Clark Kent, Batman could attack Clark. Batman (in disguise) could just get close to Clark on a crowded street and shoot him with kryptonite bullets, no more Superman.
Edit
I've read decades worth of Superman stories and have never seen him examine the air before blowing out a fire. Batman would wear a disguise (so he doesn't stand out), Superman only does a precursory scan before putting out a fire. It would not be enough to penetrate Batman in disguise.
As Clark Kent, he doesn't scan the crowds looking for people trying to kill Superman. It isn't something he would expect. Batman doesn't start sweating or have an increased heart beat due to this type of stress. He has trained himself not to do these things. Batman has gone undercover and used disguises before (Matches Malone), it's not something new to him. You don't see it much these days, but, Batman is a master of disguises.
The trick is to cause a distraction and attack Superman when he isn't expecting it. You don't fight Superman fair, like a gunfight in the middle of the street (something Lex Luthor usually does).