Danish transit in Beijing on way to Russia

Hi,

I live in the US and hold a Danish passport / I am Danish citizen. I am traveling to Siberia and Chukotka (Russia) in a few days via Beijing. Time between flights while in PEK is 13 hours. I have my tickets to fly onto Russia + I have my visa to enter Russia. Will I be allowed to go out of airport to visit Beijing City in the 13 hours I am there?

If yes, I assume it is on a Transit 24-hour permit.. anything I should know to be prepared for to obtain this?

(PS I did actually try for a visa but my travel plans were just made last weekend, and I was informed that China have announced they will not rush process passports of European citizens, so out of luck on that one.)