All of these are sedimentary rocks, which form by the accumulation of particles, or sediment, that is predominately from a single source, indicated in the name. The sources are often classified according to their chemical composition or their size; for example, sand is typically distinguished from silt or mud by the size of the sand particles, and limestone is distinguished by being composed of calcium carbonate from animal bodies.
There is no uniform way of saying that X rock will always be older or younger than Y rock. The Law of Superposition tells us that rocks on the top of a strata will be younger than those on the bottom, but we cannot take two sandstone rocks from vastly different strata and say that both have a certain age in common just because they're sandstone. What we can say is that it takes these rocks a different amount of time to form, based on their composition; for example, it might take more time for sandstone to form because its particles are so large, and therefore harder for the earth's heat and pressure to mash together into a rock.
The process of turning into a rock is called lithification or diagenesis. This is a result of several different factors, such as temperature, pressure and depth, the surrounding environment, and the size of the particles. There is no uniform rule we can use here to say that one rock will always be older or younger, but we can apply a few common-sense principles; for example, the finest-grained particles, clay, will be easier to cement because there is so little space between the particles, leaving no room for water or air to keep them separate. However, a more diverse specimen like limestone, which doesn't have uniform particle sizes, would not be subject to this sort of prediction.
In general, I would assume that mudstone and shale would lithify before sandstone, because of their smaller particle sizes, whereas limestone would not be classifiable without knowing more about its depositional environment.