Proving that co-finite languages can be decided in constant time
Your observation that co-finite languages are regular and hence in linear time is absolutely correct. Unfortunately, it's not enough, since there are definitely regular languages that require linear, and not constant, time (e.g., even checking if your input matches&nbsp;$0^*$ requires you to look at the whole input, unless $\Sigma=\{0\}$).

There are only finitely many strings not in a co-finite language. So, how much input do you need to read before you know whether the input is one of those?