A lot of Xbox Live games have this issue when they first come out. They're not able to connect until a week or so after you can download the game. The same thing happened most recently with (the x86/x64 only) Pac Man Championship Edition.
Edit: I take it back, Pac man just came out for ARM / Windows RT this evening.
This is a common thing with new Xbox Live Games. After a few days it will work. The longest I had the problem with was Pac-Man. Ended up emailing the company and they resolved it basically the next day. Great customer service!
You'll see in a few days that you're wrong. PacMan took about a week before it was fixed server-side. Fruit Ninja took more like 3 days. It's happened with every single Xbox Live release at their initial launch. The reason you don't notice it usually is because you don't actually see games in the Store immediately; they only show in new releases a few days after launch, despite being present (and searchable) in the Store. Because you already had Cut the Rope installed, you got the update to the Live version as soon as it was approved in the Store, in this case.
You don't have to believe me, but it's been the same way for every other Xbox Live game.
Edit: And... I was incorrect in this case- see below.
It has nothing to do with buying it on your phone. Microsoft has no method in place for that.
It's because you downloaded it when it was free. So basically, the original payment structure was getting it free, and then paying $5 as an in-app purchase to get the rest of the levels.
It appears with the switch to Xbox Live that they switched that payment structure to be $5 up front. So, if you already had downloaded it for free, you automatically get access to the full version because there's no differentiation between downloading a free app, and buying a paid app. It's kind of like when you download a free app that you 'buy it' for $0. So since it's already on your purchase history, you get free access to download it at a later date. Basically you just got lucky in this one specific case.
Manually check for updates; open the store, swipe from the right, select settings, and then app updates, and check for updates. That generally gives you updates a few hours before you get the notification on your tile.