How can the Universe be infinite if it was localized in a microscopic volume just after the Big Bang?
Well, this is absolutely right: if the Universe was localized in a small volume just after the Big Bang, it can have expanded to a large finite volume now but not to an infinite volume. But, the misconception lies in the mental image (carried by many, including physicists) of associating the Big Bang with a universe localized in a finite volume. Even an infinite Universe can undergo a Big Bang: each microscopic finite volume in this infinite Universe just after the Big Bang will eventually expand to a very large volume.
So, one should be cautious to make the distinction between the full Universe and the observable Universe (the portion of the Universe that we can observe, typically 14 billion light-years, since 14 billion years is the age of the Universe). The boundary of this observable Universe can be understood as an horizon.
Quite often, people say that they have difficulties picturing an infinite Universe. But, isn’t it more difficult to picture a finite Universe? What happens when we reach the boundary (unless the Universe is like a sphere)? Well, the notion of horizon should help you in this case: the observable Universe has a boundary, the horizon, but beyond the horizon begin the unchartered territories of the (infinite) Universe.