How do I complete Jimmy's sidequest?

You must set Jimmy as your active buddy and then head to the farm. If Jimmy is your active buddy, the Rancher will be there, and after some dialogue he'll give you the key to the barn.

Unless you can soak a lot of damage or have the means to take out a couple enemies in a single turn, I highly recommend keeping Jimmy as your buddy for the fight in the barn, as his ability to put enemies to sleep is highly useful in against the group inside.

After that, just climb the ladder and retrieve the flute from a chest on the upper level.