Paul Lambert's men hover perilously above the drop-zone after taking just nine points from their opening 11 games.
But they produced a spirited display against City's neighbours Manchester United on Saturday, leading 2-0 before finally succumbing to Javier Hernandez's late winner in a 3-2 home defeat.
Weimann, who netted both goals against United, believes Villa's performance shows they can return from the Etihad Stadium on Saturday with points in the bag.
Villa have already beaten City on their own patch this season, with a 4-2 success in the League Cup third round.
Austrian international Weimann, 21, said: "We don't have fear, and we shouldn't have. The last few weeks we've got confidence because we've won a couple of games.
"You've seen the way we played against Manchester United. At the moment we think we can take on anyone and at Manchester City we'll be the same.
"We went there in the cup and got a win. It's going to be a really hard game but after this performance we'll go there with a lot of confidence."
He added: "I'm sure that if we play like that we've got a chance of getting something.
"The gaffer said after the United game if we keep playing like that we'll get results, even against the top teams."