“Wrap my arms around his neck” is most natural. The other version is grammatically OK, but less natural sounding and longer than necessary.

Both are weird but the second is wrong. I wrapped my arms around him. I wrapped my arms around his body. I wrapped my arms around his back. I really don’t know what “wrapping my arms around someone’s neck” is. You could say “I put my hands around his neck”. That might mean you were intending to choke the person. You could say “I put my arms around his neck”

It's more common to hear " wrap my hands around his neck."