Easter is a wonderful time to be in Athens, it’s the most important holiday in the Orthodox calendar. You should go to any church on Good Friday and see the service, they have the holy bier all decorated with flowers and take it round the parish while the priest and cantors sing hymns, all the people hold candles, you are always welcome to join in the procession.
On Saturday night about 11:30 pm everyone will be in church there you can hear the ester “resurection” service, at midnight the lights go out and one candle is lit from the eternal light at Jerusalem and this is then passed round till everyone has a lit candle. Everyone says to their neighbor “christos anesti” Christ has risen. After the end of the ceremony many “epitaphii” with lots of flowers are being walked around the city. I suggest Plaka area because it is a no cars area and you will also enjoy the walking, the people and the mild weather.
There is not some kind of dress code about Easter ceremonies. It is used to dress a little bit conservative but this just mean: avoid shorts and Hawaiian-like colorful shirts for men and avoid mini skirts high over the knee and big cloths openings for women. It is not a matter of not being allowed somewhere, but a matter of not feeling awkward.
Local people usually wear costume and tie, especially on Saturday evening, but under-25 locals dress use more casual cloths. I think for men, khakis or a nice not worn pair of jeans with a simple colored shirt without a tie and a light jacket is appropriate for both days. Simple and elegant. As for women jeans is a no, no but pants is not something strange. Skirt is not an obligatory item. A nice cloth pair of pants with a shirt or a pullover is ok.
All these are different if attending in an isolated monastery. Dressing code for women is really different. Pants and near the knee skirts are a no no. Prefer elegant long skirts with a flat pair of shoes and you are ok.
Anyone is welcome to the serives, you don’t have to be Greek or Christian to go. It’s a wonderful way to understand Greek society.
And last, you must remember everything will also be closed on Monday as that’s a holiday too.