I was in Finale in October. Awesome spot. Amazing food and climbing and be sure to check out the beaches up by Capo Noli (the one at Finale Marina with the boardwalk is not nearly as nice).
To my knowledge I've never heard of anyone renting ropes anywhere, except maybe at a climbing gym for indoor use only. The town of Finalborgo (just up valley from Finale Ligure) has a ton of climbing/gear shops, as it caters to outdoor enthusiasts from all over Europe (mostly Germans/Austrians/Swiss and lots of mtn bikers). You could probably score a pretty good deal on a new rope at one of those shops, but that's probably your best option, other than just bringing your own.
As for climbing beta - you need a car to get around there as the crags are spread out around a couple valleys with narrow curvy and steep roads. I bought the Finale Climbing guide by Marco Tomassini and it was pretty good at getting us to the crags. Definitely don't miss the climbing at Grotta della Edera, Rocca Degli Uccelli and Grotta della Strapatente. Also, for what its worth, the grading is pretty stout compared to US sport climbing and the rock is slick as shit there. Still super fun regardless!
I plan to be in the area from late June on into July and expect to have car and gear but no partner.... so anyone looking at that time, keep this thread in mind!
It may be hot but morning starts should be OK. And if it's too much, other areas North and East are not far.
I've been to the area a few times before but was not able to climb. SCherry is right about the food. The hiking in the hills is great too and the mountain villages make great trips.