Zoe -- €80 in Mid-June for a prime location is a BIG challenge -- especially when you want to be very close to landmark sites, because when a 2 year old has a "melt-down" you want to be near your hotel. AND the best bargains in best locations already are almost full, despite the whole Euro headlines. My primary advice is to look SOUTH of the Acropolis not only because it's close to the key sites, but because it's quieter, and also has the wonderful pedestrian promenade all along that side, and also near to the National Gardens, which has playgrounds.
Booking.com has a FAB "Sort" function: choose "landmarks" as the category, and in that list, pick "Acropolis." The list of hotels will all be within a mile of Acropolis . thus automatically will be in "nice neighborhoods" pretty much Avoid any locales listed as Omnia (grungy)or Psirri (latter too noisy). Here's that Booking Link: http://www.booking.com/landmark/gr/acropolis.html?sid=71fb054d34e3520d4a87a9657074d694;dcid=1 Then you sort by price for doubles (they can add a cot for the tot). I just now did a "dummy booking" of 5 days June 15-20 -- so your limit would be €400. There are NOT many under your limit --- Hotel Nefeli and Acropolis House (both are shabby but in Very good locations, and clean). Hotel ATHOS is €490 for 5 nights All of these have only 1 room left! And a Look at the MAP of the Plaka area shows that almost ALL the hotels anywhere within 3/4 mile of Acropolis -- high-rate as well as budget -- are SOLD OUT!!! for those dates. (including my darling budget Gem Hotel Phaedra, which I booked in December for early June)
ZOE ZOE -- drop everything. Right NOW. and try to book one of these. Do not dither!! If you can get one of these, rejoice! If not, you either face grungier, farther away, or a price jump to well over €100 per night. Act NOW! ... and good luck!! If you are faced with other less-good choice get back here pronto & I"ll try to advise.