A construction apparatus and method for lifting structure integrally over obstacles in vertical direction to a position with different horizontal projection. The apparatus has two platform beams placed each other in parallel, two lifting beams laterally relative to the platform beams and a lifter equipped with the lifting beams. Push instrument kits are installed on the platform beams and the lifting beams respectively, and a plurality of sets of baffles installed on both sides of the platform beams to provide the push instrument kits with a counter-force when the push instrument kits work. When the structure encounters obstacles during the process of lifting the structure by two lifters, the posture of the structure can be altered by operating the push instrument kits and the lifters, such that the horizontal projection of the structure will be reduced and the structure can therefore be lifted over the obstacle in vertical direction.