It will be no problem to get a taxi at midnight (or even later) at the airport. As suggested above just make sure you have the hotel address in Chinese, together with the telephone number and a working mobile phone (though the driver will call the hotel using his/her phone if needed to get directions).

I live over on the north-west of Beijing and the fare to/from the airport is always somewhere between 95 and 115 RMB (including the 10 RMB toll fee) depending on the traffic and time of day. The price per km does go up a bit at night (I can never remember exactly what time this kicks in) but I would guess that at around midnight the fare to anywhere on the east-side should be under 100RMB as suggested above.

If you are arriving at T3, just exit customs, head for the elevator and take it down to the lowest floor (either marked as B1 or B2 - it's marked in English in any case on the button in the elevator), walk out and the taxi stands are directly ahead of you. I haven't arrived at T2 for a while (or at T1 for even longer ...) but as far as I recall the taxi ranks are on the same level as the arrivals.