Your three days will be very busy seeing the Acropolis, the new Acropolis Museum, the Agora, and the Plaka, the National Archeological Museum, a hike up Lycabettus Hill. Depending upon how slow or fast you like to tour, you might have time to add the Benaki Museum or Anafiotika, the village on the northeast side of the Acropolis. I can't recommend hotels in your price range. If you can go a little higher, a good spot right near Monastiraki Square is Hotel Attalos, However, keep looking and you'll find the right price.

Here are some good hotels that are close to your price range :

Hermes - Herodion ( in the Acropolis area)
Amalia ( opposite National garden) - Parthenon ( a few steps from the New Acropolis Museum and the Acropolis red line metro station) - The Athenian Callirhoe ( close to the Zeus Temple and the Acropolis metro station)

As for what to do off the beaten track

1. Take the tram from the Syntagma Square to Glyfada, a posh southern suburb on the sea, with endless shops, cafes and restaurants

2. Take the green metro line to the end station of Kifisia, a very green northern suburb of Athens, ideal for shopping, people watching, and having lunch/dinner in one of the great restaurants

3. Take the red metro line to Keramikos, a fun area of Athens, full of life with cafes and restaurants.

There are thousands of restaurants in Athens, personally ( like most locals) i never go to the Plaka restaurants as they offer "touristy" food.

When in the center try
Trapezaria near the Acropolis with excellent modern Green cuisine

Mani Mani in Koukaki ( near Acropolis) with innovative local cuisine of the Mani peninsula ( Peloponnese)

Melilotos in Monastiraki - excellent home cooked like greek cuisine

Attikos Greek House ( near Acropolis) with traditional greek cuisine and one of the best views of the Acropolis on the roof terrace.

Thanks the tips, Parosblue. Clausar, thorough as usual. Thank you!

Have dinner or lunch at a restaurant with an Acropolis view such as Strofi, Filistron, etc. It will cost more but its a memorable experience.
There are a number of outdoor dinner and drink venues if you exit the Thission metro station towards the walkway around the Acropolis. The side streets are also full of cafes full of young people and this might appeal to you. They put out tables with candles at night and its very festive.
If you are into museums, the Benaki, Byzantine and Cycladic musuems are close to each other and are excellent.

Yes, the Plaka area is where you want to be! We recently stayed at Hotel Plaka for 75E per night and loved it. However, we were there in Feb, so summer rates may be higher. We loved their rooftop deck. And I think 3 days is fine for a "highlights" trip.

Sorry, I've only just seen this post as I've been away in Athens for 8 days & only had brief looks at anything on the i'net
For hotels take a look at my reviews on Trip Advisor, there should be around 17 of them on there & the one I've just stayed at will appear soon (Athens Centre Square). I have never paid more than your budget for any of them, it varies quite a bit as to which comes up the best deal at the time, but check their own sites, booking.com, venere,com etc.
Any that do fall in your budget range for your dates feel free to ask me for more detail if needed.
http://www.tripadvisor.co.uk/members-reviews/Yiassas

A hike up Filopappou hill is also a great option. Ruins at the top and along the way, stunning views of the Acropolis from the top!