I travel with a guitar, both in hard and soft cases. I'm fortunate that as a DM I board early and the FA's have always put my case in a closet. I've had people board after me who were told that they had to put it in the overheads so I'd agree that early boarding is highly recommended.

I did see a hard case on Amazon for $40 which would probably suffice if it had to be gate checked. I think the FA's are also more likely to accept it for a closet is if its in a hard case, expecially since it's a small instrument.