I'm pretty sure the oldest is the Revell Iowa-class battleship, which Revell of Germany, at least, is still selling as the U.S.S. Missouri. It was originally released in 1953 - five years before the Arizona. (My source, as usual, is Dr. Thomas Graham's fine book, Remembering Revell Model Kits.)
The Iowa-class kit...well, looks like a 1953 kit. The detail level, by modern standards, is awful. (It was in fact pretty bad by the standards of 1958, when the Arizona came along.) The shape of the hull bears scarcely any resemblance to reality. (I think that's at least partially due to the fact that in 1953 the underwater hull lines were still classified.)
My recollection is that in the early fifties this kit sold for $2.00. (Maybe a bit less.) The current price at Squadron Mail Order is $17.50.
Some people regard this kit with great nostalgia, and are grateful to Revell for keeping it on the market. Other people think that selling a 60-year-old kit in a new box for a price like that is a thoroughly disreputable marketing stunt.
There's room for argument, but I think the title for "oldest plastic kit" goes to the old Frog Penguin aircraft kits, which were released in England in the late thirties. The biplane model that King George V and his speech therapist are handling in the movie "The King's Speech" looks like it could be a Frog Penguin. (Great movie, by the way.)
The first American plastic kits probably were the little 50-cent kits from Hawk. Round 2 Models has reissued some Hawk kits recently, but none of the really old ones. They date from the late 40s, before Revell existed. I don't think Testor's offers any old Hawk kits any more - but I won't be the least surprised if some of them turn up again. The 1/48 Spad and Nieuport 17, in particular, were remarkably nice kits for their era.
Youth, talent, hard work, and enthusiasm are no match for old age and treachery.