I suspect that using an inexpensive fence board type material would be better in the long run because plywood tends to disintegrate in contact with water unless it's that expensive "marine grade" kind, and I don't see much point in paying for that. The glue just gives out and the wood delaminates.
You could use whatever on the bottom of the box, but I would do it in slats spaced about 1/4" apart so that you can get some drainage. I typically use some 1x lumber that is only about 3/4" thick and 5 inches wide or so. If you can find it you can use a 1" thickness strips on the slats that form the two ends, that will give you a slight foot that holds the entire box off the bench. Important if the bench is wood, or it will rot quicker.
Line the inside of the box with plastic mesh. You could use the bulk roll
Drainage Screen or just get hold of some of that plastic mesh fabric from the craft store. They use it for latchhook or some kind of similar craft. I staple it in strips usually, but this year I have taken to covering the entire bottom of the box. Stay away from metal screen or hardware cloth. When roots grow through it it's a mess to clean up.
This pic from the Growing Box thread in Tips & Techniques will give you some idea:
One part of the picure that might be deceiving is that the bottom slats are a bit wider than they look because the ones on the end extend under the sides.
Lastly the Bald Cypress will tolerate just about any potting mix. I wouldn't necessarily do this to my pines, but you could get away with supersoil or something like that in a growing box for bald cypress, considering the muck that they grow in naturally, potting soil would be like nirvana. In general I do use bonsai soil in the growing boxes, and if the boxes are big, I might use a portion of soil that I had reclaimed from potting other trees after sifting out the fines.
Regards,
Matt