Nah, the memes weren't really present until after he started acting.
A lot of wrestlers try to branch out because the business of wrestling, and the performance itself are brutal. So as soon as anyone gets a little fame out of it, they'll start looking for endorsement deals, acting, whatever. In fact, the wwe has its own movie thing, and that's where Cena's first movie was.
Cena was pretty damn popular in wrestling before the memes. He debuted in 2000, iirc, and his first movie was out in 2006. The Marine had been filming well before he got to be a big meme outside of wrestling.
Now, I firmly believe the memes helped him gain awareness outside of wrestling, which allowed him to transition to bigger and better roles, but he's also been pretty smart about what roles he took in movies and TV, building success among fans, and building his acting to a more acceptable level.
I can't say he's a great actor, he's limited in range, and kinda lacks dramatic chops that could make him an actor first. But he's also not bad. He has steadily worked at it, improving a lot.
So I would have to say that the memes only boosted him after he already was on his way.
Tbh, his turn as Peacemaker has been his best work so far, and he was flirting with some good dramatic work with it. He's still better at the comedy and action than serious acting, but he might be able to get there in a way other wrestlers haven't because they can't. Wouldn't bet on him developing enough craft to be considered a great actor, but he's got a certain degree of nuance to him that surprised the hell out of me. If he works at it, he might end up doing something truly lasting as an actor.
I wouldn't give the credit to the memes, is what I'm getting at. He's worked at it steadily.