1. Every attribute of the universe we have observed is finite.
I'm thinking specifically of the limitations that are exemplified by the laws of nature, in particular. Gravity is limited by the force of the expansion of the universe, for example. It's not as if gravity, electromagnetism, and the strong and weak atomic forces are unlimited; one force may be imposed upon the other, in which case one of them is limited. But, whatever is limited is finite (by definition). Hence, the universe's attributes, especially those of the laws of nature, are finite.
2. The universe's past is an attribute of its temporal duration.
Unless, of course, one is an amazingly strong anti-realist, I doubt even the most fervent Naturalist will doubt premise (2). From these two premises, however, it follows that:
3. Therefore, the universe's past is most likely finite.
(3) is the equivalent to the second premise of the Kalam Cosmological Argument (KCA), which states that the universe began to exist. I like the above argument, since it isn't dependent on the changing field of astrophysics. Thankfully, the current scientific evidence confirms the universe's finitude. It is reassuring, however, that we have additional, philosophical arguments that the universe began to exist.