Time being infinite in the negative direction just means that for every instant t there is a previous instant t’. It doesn’t imply that anything has to “do” an infinite task.
As I noted in that same comments thread: it does not take time for "now" to be reached, any more than it takes distance for “here” to be reached. We may posit an infinite space without supposing anything to cross it. Similarly, we may posit an infinite temporal dimension without supposing anything (the “moving ‘now’”?) to have traversed that.
The problem lies in conceiving of "the passage of time" as being a kind of movement. We imagine the present-marker "starting" at the beginning of time, and moving forward into the future. But this picture belies a deep incoherence. It takes a second dimension - time - to move along some dimension. (Think of a graph plotting the change in y-axial distance against the x-axis of time.) But what is the present-marker moving through, as we track its changing temporal location? It can't be moving through the first-order timeline, since that is rather what it is moving along. We need to posit another temporal dimension, a 'meta-time', in which it can traverse first-order 'time'. This leads to infinite regress, and an absurd commitment to infinitely many temporal dimensions.
We must conclude that there is no present-marker, or "moving 'now'". All times are on an equal ontological footing, the same way that all distances are. 'Now' is no more a privileged time (or "one true present") than 'here' is a privileged location or "one true place". The significance is merely indexical. Now is the time I'm at, and here is the place. But there are other places and times, no less real and existing than my own.
To avoid regress, we must recognize that time exists atemporally. Each moment stands in temporal relations (e.g. "before" or "after") to other moments, and indexically represents itself as 'present'. And that's just how it is, eternally. The moments themselves don't change. Rather, "change" is merely the fact of one moment differing from those which stand in the 'before' relation to it.
Besides which, even if they were possible, any external changes to time itself could have no impact on our experiences. At this moment in April, I have memories of March, which I conceive of as "last month". But if God were to rewind time, to “replay” the universe backwards, we couldn't tell the difference! God would see April first, but what he would see is me with my memories of March (and of course no memory of the "future" moments which God had previously shifted the present-marker from). He would later see those memories form, back in March. But the order in which God views things, from "outside time" (so to speak), cannot affect the experiences we have within our chronology.
Consider the movie Memento, where the viewed chronology is all jumbled up. The first scene we see is the last event to occur within the movie's chronology. As the movie progresses, we get to see earlier events. The fact that our external view is jumbled up doesn't affect the true chronology of events internal to the movie. Those events stand in 'before' and 'after' relations simply in themselves, quite independently of our viewing the movie.
The idea of the "moving 'now'" or present-marker is like the external movie projector. It highlights a sequence of moments, one after the other. But it needn't show those moments in the correct sequence (as represented internally by those moments themselves). So even if there could be a "moving 'now'", it would be strictly irrelevant to the proper chronology of things. April occurs after March, no matter what order the moving 'now' might happen to show them in. The 'now' could move backwards through time, but it would make no difference -- either to our experiences, or to the first-order temporal facts.
Since it makes no difference anyway, we clearly don't need the moving 'now' at all, any more than a set of movie frames needs a projector in order for there to be facts about the movie's chronology. It is enough for the frames, or moments, to exist. They internally represent their own chronology. No external highlighting is needed.
We could imagine a movie with infinitely many frames. Perhaps we couldn't watch it from start to finish, just like a moving 'now' couldn't traverse an infinite past to reach the present. (Aside: this might actually be possible via hyper-acceleration, however!) But in neither case is such external highlighting necessary. Each frame/moment exists, and represents itself as 'present', and as 'after' other moments. And that's all we need. It's perfectly coherent to recognize that there could be a moment -- perhaps this one -- to which infinitely many other moments stand in the 'before' relation. That is, we could have an infinite past.