God's attributes are traditionally divided into two groups, at least in the theological circles I'm familiar with. He has communicable attributes, those he shares with us, and incommunicable attributes, those that belong to God alone. This system of categorizing God attributes isn’t perfect. None of God’s attributes are shared completely with us, even the ones we call communicable. And since we are made in God’s image, even his incommunicable attributes have slight human parallels. But this system, imperfect as it is, is the most common way of classifying the divine attributes, and it’s still a useful way of looking at them.
But omnipresence doesn't just mean God exists everywhere. It means he is not contained in space. “But will God indeed dwell on the earth?” King Solomon prayed. “Behold, heaven and the highest heaven cannot contain you; how much less this house that I have built” (1 Kings 8:27). The universe can’t hold God, not so much because it’s not big enough to hold him, but because “big enough” doesn’t apply to him. He is not spatial. In eternity past, without creation and without space (which was created by God when he created the world), God was everything he is now, omnipresence included. The universe can’t contain God because he is pure spirit. He has no physical dimensions.
1] Spurgeon, Charles, “God’s Nearness to Us,” Christian Classics Ethereal Library, http://www.ccel.org/ccel/spurgeon/sermons33.xxxvi.html. Spurgeon was paraphrasing a quote from Blaise Pascal: “Nature is an infinite sphere in which the center is everywhere, the circumference is nowhere.”