Not quite. You have the English question form or "when", which is expressed with cén t-am, cén uair, cá huair, or (for your barbarous Munster speakers!) cathain. Then you have the other form of when (Think "when I run", "We will leave when he comes"), which corresponds to nuair a.
I eat when I run simply implies that when running happens, eating happens (but not necessarily at the same time). For instance, perhaps you mean you eat to take in carbs before your run or protein afterward, but that you eat tied to your workout. "I eat while I run" would imply that you are running with food in your hand, taking bites as you step.
The a is needed because the conjunction nuair is followed by a relative clause, and a is the particle that introduces an affirmative present-tense relative clause. (“I eat when I don’t run” would be Ithim nuair nach rithim ; the a would normally cause lenition on the following verb, and the nach would normally cause eclipsis on it, but since rithim begins with an R, neither mutation happens.)
I agree, but maybe there is more to it: I've read the comment below where "Antaine1916" says that "nuair a" implies a kind of conditional relation between the two actions. Could we really interpret "nuair" the same way we might the English "when", expressing that "eating" necessarily happens each time I "run"? Edit: To add up, I've just had a sentence "Ma rithim, ithim.", so I may reformulate my question by asking: are these two sentences synonymous?