I have a Philippine Passport with US Greencard ( US Resident Card) booked flight to Manila

The first and the most important detail is that you are eligible for 24 hour visa free transit and you don't need any kind of visa.

Anyway, to be allowed to exit the airport, you will have to apply for a temporary entry permit at 24 hour visa free transit counter upon your landing in Beijing. Here you have to be aware that the issuing of this permit depends entirely upon discretion of immigration authorities on the day of your arrival in Beijing. In other words, you could be granted a permit, but also be denied. It all depends on the policy of the immigration authorities on the day of your arrival. Also, if you want to empower your application, you can book a tour in advance and have a confirmation in hard copy that you will enclose with your application.

It's a bit like a roulette.