Coffee beers do contain a bit of caffeine, though the exact amount varies depending on the beer. Caffeine Informer estimates the average coffee beer has less than 50mg per 12 fl oz. Given that average, you would have to drink around six coffee beers to get the caffeine buzz of one cup of coffee.
We don't recommend drinking day-old coffee, especially if it has gone rancid and has accumulated an unpleasant smell and/or taste. Brewed coffee also has a tendency to accumulate molds especially when kept outside the fridge. Do not drink day-old coffee if it has milk mixed in it, unless you kept it in the fridge.