If a stock pays a dividend to stockholders that own the stock on a specified date (ex date), but that stock holder bought the stock and only held it for an hour would he/she still get the dividend paid?

If you hold the stock at close of business on the dividend declaration date, you get the dividend. So you only have to hold the stock for one minute, and then you can sell it first thing the next morning if you want. The "ex-dividend date" is actually the first business day after that - and is important because all else being equal it is expected that the market value of the stock declines by the dividend amount.