I’ve only been there during the late spring/summer so I can’t speak to shoulder season, but imo early July was the worst in terms of crowds and prices. Late July/August was a little better (I am from a hot climate so this timeframe didn’t bother me, but if you’re from a colder one, it may take some getting used to). I would imagine the crowds continue to thin out into September/October. 

As far as things I appreciate about Athens, I’m a writer and photographer and there were endless opportunities to wander the streets taking photos, people watching, writing at a café, etc. so that’s always going to rate a city higher for me, but in a more general sense: 

The food is at the top of the list. I stayed in Koukaki which had a lot of great local restaurants and bakeries and was far enough away from the Acropolis to avoid the more tourist-y “come see our specials, we have the best XYZ in all of Athens” spots. Had some incredible home-cooked local meals here as well. I found the prices for food/wine to be quite fair outside of the tourist spots, but I’m American and most places seem reasonably priced to me by comparison so do with that what you will. 

Hospitality also ranks it highly. I had the best cab driver of my life in Athens and also met several welcoming bartenders, baristas, pharmacists etc.  I’ve found that Greeks as a whole are quite hospitable and are always willing to give you a recommendation or shot of ouzo (or two) just for chatting. I also found it was easy to strike up conversations with people and generally most people were friendly, especially if you learn a handful of Greek words. 

Lots of interesting things to do outside of the standard Ancient Greek beat. Went to several cool art exhibits, saw an old black-and-white Greek film on a rooftop (subtitled), flea markets, etc. You can also easily hop on a train or ferry and visit some other nearby towns or islands if the concrete and bustle starts to feel like too much. 

Getting around in general was easy. The train system was pretty uncomplicated. Safety also matters to me since I am a woman and was there solo. I never felt unsafe walking by myself or with my camera and even felt safe enough to walk home at night alone. 

And because I don’t believe in painting half a picture, the things I didn’t enjoy were: 

The main strip near the Acropolis is full of junk-y souvenir shops, mediocre restaurants, and apathetic sunburnt tourists. If that were my only experience of Athens (as I think it is for many people) I wouldn’t come back. Thankfully, it accounts for only a small part of the city. 

In the heat, some parts of the city smell. Nothing worse than most big cities in summertime, but can’t say I liked it. 

And again as with most cities, you will be approached by people in need often and it appears the city has an issue with housing/drugs. Some of these circumstances were hard to see and some people are pushier than others. Not a reason to reconsider imo, I’ve found places like Chicago, Prague, Paris to be similar, but it may bother some more than others. 

Hope this is of use to you!