There are more things to be done: shoot the shootable switch, use the rope, there are some climbable walls...
I don't know where you are. Maybe you need to lit a torch? I'm not sure if you're there where I think you are. The torches are behind some pushable block outside in the area of two fountains. Can you post a picture?
Pull the switch, run, grab, pull up, run, jump, grab the ceiling, monkey swing to the door, run through the door. You need to be fast. When you think, you're fast enouh, you may save between your time run. If you loose some seconds, you may reload in between the timed run. That is how I do all the timed runs. I hope you are not trying to climb onto that pillar. The top of the pillar has spikes, you must run around to the right, across the platforms and monkey climb.