Am I yotzeh if I (a non-Jerusalemite) give it to an organization in Jerusalem who gives it out on their Purim since a mitzvas Hayom on my purim?
When giving Matanos L'Evyonim, the money should be given on Purim and received on Purim. For one who wishes to give Matanos L'Evyonim to Aniyim in Yerushalayim this presents a problem. Since they celebrate Purim on a different day, how can we give them Matanos L'Evyonim and still be Mekayem the Mitzva? If we give it on the fourteenth when we observe Purim, it is not Purim for the recipients. If we would wait until the next day, when it is Purim for them, it is not Purim for us. The solution, (Rabbi Yosef Zimbal, Moadim U'zmanim, Vol. 1, No.1.) is that we can give money to a Shaliach on the fourteenth and that this would be distributed on the fifteenth when it is Purim for the Aniyim.
As long as your giving happened on your Purim, you fulfilled your obligation. The halacha is that if you wake up Purim morning and look around town and can't find any poor people, you can send the money in the mail, or just set aside the money for whenever you find a poor person. Chayei Adam 154:28, among others.
ואם הוא במקום שאין עניים שם, יכול לעכב מעות פורים עד שיבואו עניים, או שישלחם למקום שירצה
I suppose it might be slightly more ideal to have it distributed on your Purim too; how you weigh that against the preference for the poor of Jerusalem over, say, Tel Aviv, I don't know. But either way, you fulfilled your obligation.