When we went to Istanbul two years ago I do not remember any of the many travel agencies advertising day trips to Athens. You'd need to go through immigration both ways I think which means you'd spend most of the day in the airport. Also not sure if the typical tourist visa for turkey allows you to leave and then renter. You'd want to check that.
I have spent many days in the tourist areas of Istanbul, and I have examined many travel packages offered by travel agencies. I have never seen any company offering a day trip to Athens. We also have never had anyone on the Istanbul Forum discuss a day trip to Athens. There are lots of companies that offer combined tours of Istanbul and Athens, but I do not think any of them offer anything less than 7 days.
I think you could arrange this for yourself. There are many airlines that are based in Turkey which fly to Athens, and the fares are very reasonable. You would probably have to look at the flight schedules to determine if it would even be practical for you to contemplate a day trip. If you did find a workable flight schedule you could hire a guide in Athens if you feel you need one, or perhaps join a group tour?
Like the previous poster I think you will spend a significant portion of your day going to the airport, managing security, leaving the airport, managing immigration and customs in Greece, and then repeating all of that when you return to Istanbul. It is something I would not consider because of the time involved in flying, particular between two countries.
You might email Rina at Insight Travel in Kas, Turkey. If there is anyone who could provide practical tips on how to do this, she would be the person I would consult. While she is now based in Kas, she lived for many years in Istanbul and she manages travel all over Turkey. She might be able to check airline schedules and fares if that is difficult for you to do from Boston.
Given the traditional "rivalry" between the two countries I doubt whether too many companies/guides in either country would advertise this kind of tour.
It really is not possible to do a day trip to anywhere by air. You have to be at the airport 2 hours before departure, Athens-Istanbul is around 90 minutes, and it takes at least 30 minutes to get to/from airports. So that is 9 hours spent just in travelling between the city centres and back. Not much time left for sightseeing.
I doubt anyone would do a day tour to Athens by the time you got there it would be time to return, I tried googling it but only trips with more days came up.
..although I personally wouldn't rec it, it can be doable if the flight times work out with early morning departure and evening return! Along with National Carrier, you can also check www.olympicair.com/ www.aegeanair.com/ You can sign up for email alerts with both.. the latter offers some fabulous deals throughout the year;-)
You would need to make your own arrangements for the day.. either via an organized City tour offered by numerous top operators like www.chatours.gr/ www.keytours.gr/ or make arrangements for a private tour.. you can send an email off to www.athens-walks.com/ and see what they can offer for the day;-) FWIW, if you've got the time and can add a day, you may just want to consider overnighting?;-) Best.. and enJOY!
I also do not agree that a day trip to Athens is impossible. People travel to Ephesus and back on a day trip. The only difference is that you do not have to cross borders to do this. Crossing the border twice in one day seems as if it will add more transportation time, and the regulation about being at the airport two hours before an international flight could prove time-consuming as well. The actual flying time is only about another 30 minutes.
If you could leave very early in the morning and come back very late at night it seems as if you would be able to see a bit of Athens between plane trips. It is, as I stated, something I would not consider. If you want to pursue this you have some good resources to explore. You can determine if it is a practical idea.
I also agree that you will not be bored with six full days in Istanbul. I have spent hundreds of days in Istanbul and there are still many things on my list to see and do. Every year when we walk past the travel agents in Istanbul advertising great rates on flights to Athens, we say to ourselves we ought to do that. Somehow, however, the 4-6 weeks we allow ourselves in Istanbul seem to slip away and we never fly to Athens. We are old and slow I guess.
I'll add a personal perspective, similar in some ways. I spent almost a week in Athens, but took a day and a half trip to Istanbul on the tail end of my trip. I flew over early in the morning and back the following afternoon. One thing now mentioned is weather. When you plan something like this and lock in your date, weather might change things. In my case, an unexpected snowstorm/blizzard.