is there a public transport to travel on jiaozhou bridge?what is cost? is boatride better to see it?

Mongolia and Jiaozhou Bridge in Qingdao are at two directions of Beijing. You need to visit one by one.
From Beijing to QD: there are both high speed trains and flight (you can search the schedule on this website). I personally prefer train for it costs less. Upon arrival, take a taxi to ride over the bridge.
Mongolia is a large area? which city are you going to visit?