This appears to be part of a prayer associated with Blessed Sister Faustina, a Polish nun who reported numerous instructions for prayer that she received from on high. According to Lawrence J. Gesy, The Hem of His Garment: True Stories of Healing (1996):
Sister Faustina was a young, uneducated Polish nun who in the 1930s received a message of mercy from the Lord. In obedience to her spiritual director she recorded over six hundred pages of divine revelations about God's mercy. She lived a cheerful and humble existence, suffering greatly in silence with tuberculosis. In 1938 at the age of thirty-three, Sister Faustina died. Her writings are contained in the book Divine Mercy in My Soul—Diary. In 1993 she was beatified under Pope John Paul II and is presently a candidate for canonization.
That is by way of general background. The specific sentences that the OP asks about appear (in substantially modified form) in a later section of Gesy's book, titled "The Three O'clock Hour":
In His revelations to Blessed Faustina, Jesus asked for special, daily remembrance at three o'clock, the very hour He died for us on the cross:
"At three o'clock, implore My mercy, especially for sinners; and, if only for a brief moment, immerse yourself in My Passion, particularly in My abandonment at the moment of agony. This is the hour of great mercy for the whole world. I will allow you to enter into My mortal sorrow. In this hour, I will refuse nothing to the soul that makes a request of Me in virtue of My Passion" (Diary, 1320).
So much for the context of the OP's quotation. Evidently, the person who is being quoted is Jesus, as transcribed by Blessed Sister Faustina. The plain meaning of the final sentence is that Jesus will not refuse any request suitably made in prayer during the three o'clock hour. There is no indication that time zone adjustments are necessary, either.
But note that the object of Jesus's mercy is the soul of the person submitting the request, and that the request must be submitted "in virtue of [Jesus'] Passion." That doesn't sound like a promise to tell you whether the Mets will win next Saturday so you can bet a bunch of money on them.
The wording as rendered in the OP's question seems rather prosaic and contextless by comparison:
During this hour, I will refuse nothing to the soul that makes a request of me. In this hour, you can obtain everything for yourself and for the others for the asking.
In fact, it sounds more like the offer of succor made by the gentle angelic Totò in Vittorio De Sica's Miracle in Milan, who begins miraculously granting wishes among the desperately poor people in a Hooverville-like encampment outside Milan, and quickly finds himself surrounded by escalating demands for "milione lire" and "milione milione lire" and "milione milione milione lire" and "milione milione milione milione milione milione milione milione milione—milione!"
In any event, you could look at the promise channeled through Blessed Sister Faustina as a guarantee of a material windfall—the pony you've always wanted, say—for you and your friends just by (in effect) rubbing the magic lamp in the correct way. But I think you would be closer to the spirit of the promise if you understood it to be guaranteeing that the prayers of that hour, submitted by a devout and pious believer in the (temporarily) shared experience of the Passion of Christ, will be heard by Jesus and that Jesus will surely accept (not refuse) any request such a prayer contains.