When someone says that the universe is flat it should be infinite. But, this contradicts the Big bang because infinity can never be reached from below. If the universe is in fact infinite then shouldn't it be infinite always, I mean all the 'time'? Is Big bang a lie or universe does have a curvature?
There are two misconceptions in your question:
First, flat does not necessarily imply infinite: Topologically, a torus is just a rectangle with oppposite edges identified, and thus can be as flat and finite as said rectangle. However, in the specific case of Friedmann universes, spatial flatness does indeed imply an infinite universe, so I'll let this slide.
Second (and more importantly), the Big Bang is a metric degeneracy (distances go to 0) that can happen whether or not your space is finite or infinite. It is not an explosion located at a certain point within space growing to finite size within finite times, but something that happens everywhere all at once. Therefore, the term 'everywhere stretch' has been proposed as a more descriptive name in certain videos aimed at the general public.