We did a weekend break to Athens a few years ago. We went in March and were lucky enough to get some nice spring weather but I'm sure early October will be good too. The weather should still be warm and sunny but not too hot for exploring the city. There is plenty to see, if you are interested in the history and shops to explore too. We went independantly and got a very good deal on the St George Lycabettus hotel, as it was out of season but there are plenty of places to choose from for all budgets. It's easy and cheap to get from the airport to the city centre by metro or a bus that runs 24 hours. I'd go for it.

I go to Athens 10-12 times a year, any time is good! I avoid July & August because it's just too hot for me, but on the other hand it's not so busy then as the Greeks are all on holiday. Early October should be a good chance of it being fine, a small risk of rain, but it doesn't usually last too long.
I have stayed in many different hotels in a central location, just ask if you want advice on any. Be warned that Omonia area is not the best to stay in, look in Plaka, Syntagma, Koukaki & Makrigianni areas. If you provide a budget in euros I could make a few suggestions.
From the airport the metro is €6 (or €10 for 2 tickets), the bus is €3.20 & runs 24/7.
Back there for a visit in Sept & again mid Oct!

I love Athens! October is good, it's not that hot as in August (+September) and not that rainy as sometimes in November..