I think he just had a gun because he was a mafioso capo and carried it for protection. They were going to meet with Barzini's men, so it could have reasonably be assumed he had it on him "just in case".

I doubt he was going to do the hit himself; it probably would have been Barzini's men burst in and gun Michael down, Tessio pretends to shoot them (So any witnesses could point out to the Corleones that Tessio wasn't in on it).

I think Tessio was being set up himself, anyways. Even if Michael didn't know about it being a trick, I think Barzini's guys would have "accidentally" gunned Tessio down too, since Barzini's not stupid - Tessio would still be a traitor, and a traitor can't be trusted.