Guitar case in overhead bin on flight to ZIH?

I have no experience with AA. They might have more overhead space than some other airlines.

You might have already read this on their website. (Emphasis added)

Carry-on instruments
You can travel with small musical instruments as your carry-on item on a first come, first serve basis as long as it:

Fits in the overhead bin; or
Fits under the seat in front of you

If your instrument doesn’t fit in the carry-on bag space, you can choose to buy an additional seat (LG: YIKES !) which:

Costs the applicable adult fare plus any applicable taxes or carrier-imposed fees
Is not in an exit row
Is directly next to yours

The flights to ZIH that I have been on in the past few years have been extremely full, as have been the flights of any friends/relatives who have visited with me.

Best of success in this. AA does offer two more tips to maximize likelihood for success.

To make sure you have room to store your instrument, board as soon as your group is called, or buy Priority boarding.

The size of overhead bin space can vary by fleet type and also between aircraft in a fleet. For further information on the aircraft in your itinerary, please contact Reservations.