half-life of caffeine molecules
According to Sigma-Aldrich, pure caffeine has a shelf-life of four years at room temperature, or many years at 2-8°C.

A caffeine solution can be stable for months even at moderately high temperatures.

So, essentially, your day old coffee has still all its caffeine in it, although probably it does not taste that well, but that's caffeine unrelated.

As a side note, after drinking coffee/tea/Coke/etc. it will take ~1-2 hours for blood caffeine levels to peak. Caffeine half-life in the body is ~3-6 hours.