Block universe+eternal universe= infinite universe? Think about space for comparison. There isn't a place that is the first place, with a second place next to that, and a third place further in the same direction as that, and so on. We can assign numbers to places as a coordinate system, but that's just us making a map of the underlying space, and when we do that, of course we're inclined to put "zero" somewhere significant to us. If space is infinite, that just means that for each place next* to us, there is another place next to that in the same direction, and so on, without there ever being a place beyond which there is no other place. No matter which direction we're talking about.
Likewise with time. There's this moment, and a moment after it, and a moment before it, and each of those has a moment further in the same direction, whether future or past. There's nothing special about the fact that we're counting up from zero in one direction and down from zero in the other direction; the numbers are just ones we're arbitrarily assigning to them, we picked an arbitrary time to call "zero" and count forward and backward from there. If time is infinite, that just means that for every moment in the future, there's another moment in the future of that; and for every moment in the past, there's another moment in the past of that.
So worrying about metaphysical problems with an infinite past is basically the same thing as worrying about whether there's a most-negative number. Whether a point in time is labeled with a positive or negative number is completely arbitrary. If you have no problem with there always being a place further in the same direction as another place, or a time after any other time, there should be no problem with there always being a time before any other time.
Or course it's still possible, maybe, for there to be an end to any of those series. It's just not metaphysically necessary for any "woo infinity is scary" reasons.
*Yes, I know; all of this "next" talk is glossing over the issue of continuity, and speaking as though space and time is discrete, just for ease of discourse. If you don't know what I mean in this aside, just ignore it, this is just for the math pedants.