Although I don't radically disagree with @ЯegDwight's answer, I think it's worth pointing out that I don't really endorse the idea that I have long [verbed] is "unexceptional".
In my opinion, this construction is at the very least dated and formal. I would also say that in most contexts where it is still used, it usually carries the implication I first [verbed] a long time ago, and have continuously (or repeatedly) [verbed] ever since. Thus, for example,...
1: I have long known that she would make an ideal wife (I knew that long ago, and I still know it)
2: I have long been married (where again, being married is a "continuous" state)
3: ??I have long married her (idiomatically, extremely unlikely)
...where #3 doesn't work at all well because obviously the only possible meaning is I married her a long time ago (a single "one-time" action that wouldn't/couldn't ordinarily be repeated).
To further illustrate the points I'm making here, first consider the massive drop-off in currency of I have long seen... over the past century or two. Also, have a look at the contexts where it's used. It's never used to mean I saw [had sight of something] once a long time ago. In fact, "to see" in such contexts is usually a figurative usage meaning understand, become aware of (i.e. - ever since the time when you first did it, you've been doing it continuously right up to the present moment).
In OP's exact context, obviously it's possible to see "beating" the game as a repeatable/continuous activity, but that's probably not really the intention. A more natural way of expressing what I assume OP means would be, for example,...
I [first] beat the game long ago, but I still come back from time to time to listen to this tune.