Depends on how old you are talking about. On earth, many organisms continuously deposit hard tissue of some kind, and the rate at which tissue is deposited varies annually based on the waxing and waning of seasons. Namely, in the winter/dry season (for the tropics), growth is slowed because of a scarcity of resources, whereas in the summer/wet season, growth is faster. This creates a series of annuli (i.e., "rings") in the hard tissues of many organisms, including the rings of trees, the number of lines of arrested growth in the cross-sections of long bones in most vertebrates, the shells of mollusks, the cementum of the teeth of many mammals, and the otoliths of fishes. Similar annuli form in the earwax of whales and the eye lenses of many vertebrates, which aren't hard tissues but grow in a similar way.
There's been a lot of press recently about Greenland sharks living in excess of 100 years, but the way that scientists figured that out was due to a chance occurrence. Namely, the fact that the eye lenses of Greenland sharks lacked an enrichment of radioactive isotopes after a certain point that was created by manmade nuclear testing in the 1960s, called the "bomb pulse". The researchers then backcalculated the age of the shark based on the amount of growth predating the bomb pulse to the growth after, which they could do because they knew how much had occurred since the bomb pulse. This wouldn't work for most organisms due to the way radioactive decay works, the lifespan of most organisms is so short that the change in C14 enrichment ratios from two decades ago to now is barely even noticeable.
Additionally, organisms are constantly taking in radioactive carbon isotopes through their diet, which is how it is possible to infer that an organism died so many years ago, death starts the radioisotopic clock ticking down because no new radioactive carbon is being incorporated into the organism.
What's going to work best for your aliens is going to be entirely dependent on your alien's physiology. What researchers would look for is a long-lasting, slow-growing tissue, one that is sequentially deposited and rarely if ever remodeled. For example, long bone growth rings and sometimes tree rings can be secondarily obliterated, making it difficult to back-calculate the age. Assuming a generally vertebrate-like body plan probably the first thing people would look for is the eye lens, which is pretty consistent in estimating age across Earth vertebrates. It may also work in cephalopods, but I don't think it would work in arthropod-like organisms because the eye lenses are shed and regrown with each molt. In fact, there really isn't any good way to estimate the age of lobsters and other long-lived arthropods for the exact reason that no hard tissue is retained between molts .