Prior reflects your prior knowledge (or assumptions) about the modeled phenomenon. If the quantity changes over time, choosing constant prior may not have much sense. You could use a prior that is a function of time, one example of such prior is Gaussian process prior. So yes, prior can be set over a function.
Gaussian process is a prior over functions, where the functions are to be estimated. For many cases you don't need Gaussian process. If the form of the function itself is known (likelihood), but it depends on some unknown parameters, then you may need only the priors over the parameters.