Cheap accommodation in Ayia Napa by: Helen (website owner)
I don't know Ayia Napa at all sorry but I do know that you will find it difficult to find reasonably priced accommodation in the middle of August when the place is lterally heaving with holidaymakers.
You won't be able to stay in a private apartment as they don't offer accommodation for one night only.
You can look at hotel options via this link which lists hotels from lowest to highest price - there's also a map you can click on there to see where it's located.
On the basis you aren't going to get much sleep if you stay near bar street, you may just want to stay up all night and stay further away maybe in Protaras or even Larnaca itself where you are more likely to find something cheaper for one night. Easyhotel for instance have a hotel in Larnaca.