After doing a little bit if research, I came to the conclusion that Dr. Cid never entered Giruvegan; he merely only discovered it.
There is not much knowledge to Cid's history prior to the events in Final Fantasy XII, but Cidolfus Demen Bunansa sheds some light on his past
Not much is known about Cid's life before the events of Final Fantasy XII. He fathered three sons, one named Ffamran mied Bunansa, but his other two sons and wife are never seen or mentioned in the game.
Six years prior to the start of Final Fantasy XII, Dr. Cid traveled to the Jagd Difohr on an expedition into the Mist-covered waste. It is not known whether it was a primarily exploration-driven mission, or whether he already had some idea of what he was looking for and what he would find there. Either way, he stumbled upon the Giruvegan, where he met the rogue Occuria Venat, and discovered the nature of the stone of the gods, nethicite. Venat took Cid as their pupil and taught him how to synthesize artificial nethicite, which came to be known as manufacted nethicite.
Why didn't the the Occuria try to stop Cid and Venat from leaving? The Occuria abandoned Giruvegan centeries prior to the events that take place. For that reason, they were not present to stop Cid and Venant from joining forces.
Once the center of Ivalice centuries ago, Giruvegan was ruled by the immortal Occuria. For reasons known only to them, the Occuria closed the gates to the city and let it fall into ruin. As such, not much is known about the once great city. Now, a dense Mist covers the city, and monsters wander its many corridors.
Seeing how the Occuria abandoned Giruvegan, that brings into question why Venant was there in the first place. Venat did not agree with the Occurian's points of view and believed all races should be able to write their own history.
At some point in the intervening time, one of the Occuria went rogue, disagreeing with the Occurian philosophy. Venat felt all races should have control over their own history and concocted a plan to create nethicite that was not under the control of the other Occuria, manufacted nethicite. Venat accomplished this with the aid of Dr. Cid of Archadia, with the Emperor's son Vayne Carudas Solidor aiding them by setting up the Empire's war against Rozarria and taking the shards cut by Raithwall from the locations they conquered. Enraged at Venat's heresy, Gerun and the other Occuria decide to bring forth a new Dynast King to destroy Archadia and restore the status quo they had created.