I would like to fly to Panama City and while I am there go on a boat through one of the locks. Is this possible?

hmm I don't know. I was on a cruise through the canal. ask on trip advisor if nobody else responds here...

Perfect suggestion. I got the information I needed, plus ideas for a number of other things to see and do in Panama that I had not thought of.

Yep complete are partial transit as you like

www.panamacanal.com

There are partial transits on most Saturdays, they also run on other days during the high season - the two companies that do it are: Panama Marine Adventures and Canal and Bay Tours.
Takes about 5 hours - leaves from the Amador Causeway and goes through the Miraflores Locks and the Pedro Miguel Locks, the Culebra Cut (narrowest part of the canal) and into Gatun Lake, docking at Gamboa - you will be brought back by bus.
Once a month they do a full transit - right through to the Caribbean - but these are long days and we usually don't recommend them unless someone really wants to go ocean to ocean.

Cheers,
Andrew