It's near impossible to just phone up a stranger, introduce yourself and then set up a date. Mainly because you're going to sound like a creepy fvck by calling random girls and asking them to date you. They don't know who you are, how you look, and how you act. You have no rapport with them, so you cannot fall back on their memories of you and how you made them feel.
I'd change the bet a bit and say that you're allowed to introduce yourself as a friend and that's all you can use. Even I would blow off a random chick that called me and wanted to date me.
And the only thing I can imagine is calling them up, asking for a different name(preferably a girl) then theirs, who you're obviously not going to get on the line, and then before hanging up, you tell her she has a lovely voice and ask her where she's from. Coincidental you're from the same town as her, and you could be like "Alright, this is going to sound a bit weird, but I'm *really* curious about the person behind the voice. I happen to be at the mall <date/time>, so how about we go for a cup of coffee/tea then at <place>. Just to satisfy my curiosity, and we don't like one another, we just forget this phone call and each other's number altogether. "
That's all I can make of it...