I am downsizing and must let someone go, can I use the last hired first fired rule?

A union contract may modify the precise application of the "last hired, first fired" principle, but presuming yours is a non-union organization, this is not a bad rule-of-thumb to use. It has the advantage of ensuring that you have an objective and discrimination-free reason for selecting employees to be laid off/terminated (for example, it can assist you in defending against any claim that you selected people for improper reasons, such as gender, age or race. All else being equal, it also has the advantage of potentially reducing your termination costs as your shortest-serving employees are likely to have the lesser entitlements. The disadvantage may be that you lose a more capable employee simply because he or she has shorter service. But assuming that you have no contract (or similar agreements), policy or practice which contradicts its use, the "last in, first out" principle can be used when compelled to downsize.