Say I have airline status with American Airlines and would normally be able to use this airline's lounge. This time I'm flying with Gulf Air, but the flight is in fact operated by American. Would I still be able to use the Admiral's Club while I'm at the airport?
You say you have "airline status" with AA. If you mean you normally can access the lounge when flying AA domestically or on an economy ticket, then you can use the AA lounge - a boarding pass should only be needed when accessing a Third Party Lounge.
The exact definitions given are "an Admirals Club annual, lifetime or 30-day member, Citi® / AAdvantage® Executive Card primary cardholder, or AirPass member with Admirals Club privileges" (http://www.aa.com/i18n/travelInformation/airportAmenities/lounge-access.jsp)
If you are hoping to use the lounge due to the level of Gulf Air ticket you have bought, then probably not. See the other comments/answers regarding whether it is a oneWorld ticket.
If you think you can't access the lounge, but still want to, you can always purchase a One-Day-Pass
I'm not sure what you mean by "I have airline status with American Airlines and would normally be able to use this airline's lounge," because American Airlines doesn't provide lounge access based on American Airlines status alone. They do offer lounge access to those with Sapphire and Emerald status on any other OneWorld airline, and they provide lounge access to Executive Platinum members flying on an international flight that departs or arrived that same day and is marketed and operated by a OneWorld airline. If you're on a flight operated by American but marketed by Gulf Air, you can try for lounge access, but it's doubtful you'd get it.