Do we exhale Argon (Ar) in our breath? If yes, how long it could last in the open air/atmosphere?

Yes, air is about 1% Argon. It's totally inert so we constantly inhale and exhale it. It's not radioactive or chemically reactive, so it lasts a long long time. It must only decay by collision with cosmic rays or something, so it would be around much much longer than 2,000 years. I don't remember specifically, but I've heard of probability calculations that say every breath we take contains atoms we've breathed before; a little counter-intuitive but sounded legit. So yes, I'd say it likely we occasionally breathe some Argon atoms that were breathed by Jesus.