I have a flight leaving at 06:00 AM from OTP (Bucharest Airport). Since I will spend the entire night at the airport, I would like to get some sleep airside, because there are more comfortable options.
However, Ryanair requires non-EU passengers to get their documents checked and stamped before security.
Is it possible to do this procedure in the evening (10-11 PM), and then go through security and sleep airside?
In my experience at several RyanAir airports, no.
Often it's the same staff that arrive, check people in, and then reappear at the gate to board you. It's all done as part of an operation to have fewer staff costs. So them being there the night before? Maybe if there's another RyanAir flight at the same time. But most airline in my experience also won't check you in the previous day. I've also tried a few times ;)
Presuming it's this one, this link on sleeping in airports may help on where you could at least rest your head.
Ryanair no longer enforces their "visa check" policy in practice, so you can proceed directly airside and ignore their statements about needing to go to the check in desk first. Your visa will be checked at the gate instead.
See this post for another anecdote of skipping the checks. To clarify, there is still a "visa check" at the check-in desk, however Ryanair won't deny you entry if you skip and head to the gate instead.