Hi there, this coming July I'm going to be flying into Instabul Airport at about 23:30 (11:30 PM) and I have a 7 hour layover. I'm flying Turkish coach class and have no special statuses or cards which will get me into any lounges.

I'm looking for advice on what I should do. It seems like the prime class lounges are like 65€ for 4+ hours, so I could probably go in there and try to get some sleep somewhere. I've read that the airport doesn't really have many places to comfortably nap, so I'm not sure if I'd have success outside of a lounge. I know there are hostels not that far from the airport, but by the time I pay for a taxi, a visa, it suddenly starts looking less attractive.

What would you guys advise I do?

A lounge seems to be your best choice.

THY lounge is not possible but you may try lounges other than prime class just in case you find something cheaper.
At least you will get free drinks inluding wine and nibbles.
Good luck.

See: https://www.sleepinginairports.net/europe/istanbul.htm - you do mean Ataturk?

I have used both WOW hotels which are fine but you will be too late for the shuttle and I don't know if the metro runs that late. The lounges will be cheaper in any case.