The word you're looking for is "der (Telefon)hörer". The literal translation is "the listener" but in context of telephones it was the word for the "bone".
Ich lege den Hörer auf.
I put down the receiver.
Ich nehme den Hörer ab. I pick up the receiver/the phone.
The sentence you're looking for would be
Ich gebe ihr/ihm den Hörer.
You could still say that for wireless phones that have a base station somewhere, but it sounds a bit out of place for a cell phone. There, you would just pass on the "handy" "