Disclaimer: I do not own these characters. Prince of Persia is owned by Ubisoft, and this incarnation of Aladdin is owned by Disney.
Going through the Portal
The Prince was not having the best of days. In the last few days, the palace had been taken over by sand creatures, he had nearly died in over a hundred ways, and there was a severe shortage of edible food left in the palace. On the plus side though, he had begun to master the power of the dagger of time, and there was the possibility of a romance with the captured Maharaja's daughter, Farah. The only inhibitor in that was the prince himself, after seeing that disturbing vision of Farah stealing the dagger. Either way, they had to get to the Maharaja's treacherous vizier and re-seal the hourglass. It was during these thoughts that the prince came across a door he'd never seen before.
In Agrabah
Aladdin was having some trouble of his own, the palace had been temporally been taken over by an evil sorcerer, and there was a small mutiny by the town guard. Everything seemed to be settling down now, but there seemed to be an unnatural type of sand coming from one of the buildings in town. He got the magic carpet and flew off to investigate this odd phenomenon.
In the Prince's palace
The Prince walked through the door and the next thing he knew, he was obviously in a different world. He turned around and saw the doors were already beginning to close, so he marked the walls with an hourglass symbol to remember where the door was, before he set off to explore the new environment. It seemed to be unspoiled by the sands from the hourglass, but it still was not time to relax, after all, every moment he was here was a moment that Farah was alone and unprotected. He just needed to get some food and possibly some way of carrying water. He then looked around and noticed something; the sands were seeping through the door. This place was in danger of becoming like his kingdom. He needed to warn the leader of this place what was going to happen. Then he heard a voice saying, "Who are you?"
Aladdin had never seen this man before, but he looked strong. The man said, "I am a Prince of one of the Persian territories. My name is," but that is as far as he got before he heard the screaming from the city. The Prince then ran and leaped into the alley, going from wall to wall to keep from hurting himself on the landing. He then ran at amazing speeds to find the source of the screaming.