Was it possible, through some kind of great or significant act, to become a hero and worshipped during your life and after your death? Note that I am not talking about state-cultivated cults of personality surrounding rulers but rather more local or translocal heroes who have accomplished a great task.
Yes is the answer. Unfortunately I cannot think of many examples, nor would I imagine it was a common occurrence, but one exemplar case study is that of Marathon.
After the Battle of Marathon in 490 BCE, when the smaller Athenian and Plataean force defeated the invading Persians, Marathon became a site of worship for those who fell in the battle. A mound for the dead was built, which you can still see today (although, from what I gathered from a professor, the veracity of whether the mound one can see today is the actual one has come under dispute). Pausanias, in his travel writings, tells us:
"At Marathon every night you can hear horses neighing and men fighting. No one who has expressly set himself to behold this vision has ever got any good from it, but the spirits are not wroth with such as in ignorance chance to be spectators. The Marathonians worship both those who died in the fighting, calling them heroes," 1.32.4
As you can see, cult worship of previously regular citizens. Honestly the Persian Wars are a fascinating example of historiography; I find them interesting not so much in what occured but how those events were viewed by later Greeks. The whole series of Greco-Persian conflicts from 499 BCE to 449 BCE were presented by the later Athenian Empire as an ideological argument for their imperialist control. I won't go into it because it's off-topic from your question but if you wanted to know more I could answer.