Praying for the Redemption is no different from praying for health or livelihood. We do not consider it “mixing in to G-d's business” when we ask Him to provide us with good health, livelihood or anything else we need. We pray for these things even though we believe that G-d will grant us what we need in the right time and in the right manner.
G-d created the world so that some of our needs are provided without any input on our part, but for some things, we need to turn to G-d and request them. Our sages have established the Amidah prayer, which contains eighteen blessings, to be recited three times every day. The eighteen blessings cover the gamut of human needs and desires: Wisdom, health, unity, peace. Primary among the blessings are requests for the Redemption.
The Lubavitcher Rebbe King Moshiach has mentioned on a number of occasions: "If only we would plead and cry out to G-d with sincerity, certainly Moshiach would have been here already!" When we request any of our other needs, we have no assurance that our request will be granted. After all, sometimes G-d has already decided otherwise and our prayers won't change the reality. However, the Redemption is something that G-d explicitly promised to us through His prophets. Therefore, there is no doubt that if we ask for it with sincerity, G-d will send it to us.
Our sages say that “G-d desires the prayers of the righteous.” In G-d's eyes we are all considered righteous, as the verse states: “Your nation is all tzaddikim.” It is possible that the exile has been dragged out for so long only because G-d desires our prayers. Just one more sincere prayer from one of us can be the final step that will bring the Redemption.
Talmud Yevamot 64:1. Shealot Utshuvot Chavot Yair, chapter 219. Hitvaadut 5742, vol. 2, p. 674. 5751, vol. 3, p. 119. vol. 4, p. 288