Aspirin is pretty lousy, though there are situations where it has something resembling a purpose (like when you fail a bionic install but want to keep installing bionics, to get rid of the few points of pain the failed install gave you so you’re not installing the rest at an Intelligence hit; that’s happened to me).
Really, the best thing to do with aspirin is to use it to craft better stuff. First aid kits, smoke bombs, meth, whatever. For actually killing pain, especially in a combat situation, you need real meds. Jakers mentioned oxycodone; there are others that actually work, too. Tramadol takes a long time to kick in but is powerful and lasts a long time; it’s good to take when you know you’re going to be dealing with Ow in the next few hours. Some of the better Survival texts also have recipes for how to make better-than-aspirin painkillers out of mutant poppies (you may wish to take a tramadol before going out to pick them because the game ain’t shittin’ about’em bein’ mutant poppies).