Can i apply for a 144 hour transit visa? and how much does it cost.

Actually, you are looking for 144 hour visa free transit, since there is no such a thing as "144 hour transit visa". Also, it is absolutely free.

As far as your itinerary is concerned, it is eligible if Beijing remains your only stop within mainland China and if you intend to travel from Ulaanbaatar to Beijing by plane. There is quite a number of travelers asking about the train option, but that one is, unfortunately, ineligible for this visa free transit option.