For simplicity's sake let's say there where no significant delays (like in the poem) and the hypothetical captain in question knew the way pretty well.
Assuming you're just looking for a ballpark estimate, you can use the Stanford ORBIS tool to estimate travel times. Choose Tralles -> Krane for a pretty close approximation, and you'll get an estimate of 8.6 days.
A disclaimer would be that this is based on the Roman Empire, likely over 1200 years since the Trojan War was supposed to take place. The ORBIS estimates are based on square rigged ships, which were also common in the era of the Trojan War, so there shouldn't be a massive discrepancy with that, but there could still be smaller advancements that improved travel times by the time of the Roman Empire. It's much more likely that the "cost" estimate would be wildly different, so I'm not going to bother trying to estimate that.
This answer by /u/Iphikrates, although it doesn't go all the way back to 1200 BC, gives a good rundown of ship development in ancient Greece.
You can also check out this answer by /u/XenophonTheAthenian regarding the reality & ease of sailing around the Greek Islands.