Even though the film itself was nice and had quite of a growth for rey that one duel and also a scene before are very strange. But lets start with the beginning shall we?
He is too old you say The too old I always got the feeling that it was just an excuse so that the one who was declined does not feel that bad about it. In addition to that though the older one is the more bonds one has formed already and Jedi shouldn't have bonds as those can be used to make them fall to the dark side.
The 1 scene I mentioned
The interrogation scene with Rey. Ren was able to get into her mind until he made an error and said something that made her will go strong. At this time she EASILY overpowered someone who has trained such things FOR YEARS. As we know from the prequels raw force potential says nothing (anakin lost to Dooku in Episode 2......easily) without any training. So there are 2 options for this part. Either Rey had a training and forgot about it (during her unknown childhood part before Jakuu) or she has a WAY HIGHER force potential than Ren. High enough that she is even untrained way more powerful than he is.
The duel
Rey was on the defensive and he clearly toyed with her....only for one reason (and despite his terrible wounds). He wanted to train her in the dark side and make her fall. He had her at his mercy until she closed her eyes and did what Luke did during the battle of Yavin. She calmed her mind and let the force flow instead of blocking it out. Then she beat him fully. Here we can only speculate how she could beat someone who has way more training than her with the force and the lightsaber than her. Possible is as stated above that she has training that she doesn't remember OR that she is WAY more powerful than him (would also explain how the Great leader and Ren could feel her awakening). Additionally pointing into the WAY more powerful direction is the fact that kylo tried at the beginning of THEIR duel to telekinetically call the lightsaber........and it flew but not to him but to Rey.
A bit here additionally (personal points):
In addition though is what I personally find confusing about the duel. The stormtrooper who seemingly is NOT force sensitive easily shot ren first and then attacked him. Even with his wounds Ren should have easily been able to vanquish him without even getting a scratch. Then Rey beats him easily as soon as she does not act like she wants but lets the force flow. So aside from the two possibilities above there could be another one that Ren is just not as powerful as he wants to be. That is coutnered though by his blaster freezing in the beginning of the film. So he IS powerful. And she has to be stronger as it seems. One thing we know for sure though is that Ren is NOT as powerful as Vader was (yet). As it is still his fear that he never becomes THAT powerful.